Limits of Functions (including one-sided limits) - AP Calculus AB

Card 0 of 770

Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

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Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

Compare your answer with the correct one above

Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

Compare your answer with the correct one above

Question

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Answer

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Compare your answer with the correct one above

Question

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Answer

The limit of a function as approaches a value $x_{0}$ exists if and only if the limit from the left is equal to the limit from the right; the actual value of $f(x_{0})$ is irrelevant. Since the function is piecewise-defined, we can determine whether these limits are equal by finding the limits of the individual expressions. These are both polynomials, so the limits can be calculated using straightforward substitution:

$\lim_{x\rightarrow 0 - }f(x)= \lim_{x\rightarrow 0 - } \frac{2}{x+2 } = \frac{2}{0+2 } = 1$

$\lim_{x\rightarrow 0 + }f(x)= \lim_{x\rightarrow 0 - } 2x+1 = 2(0)+1 = 1$

$\lim_{x\rightarrow 0 - }f(x) = \lim_{x\rightarrow 0 + }f(x) = 1$ , so $\lim_{x\rightarrow 0 }f(x) = 1$ .

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Limits of Functions (including one-sided limits) - AP Calculus AB

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Evaluate the limit:

To evaluate the limit, we must first determine whether the limit is right or left sided; the minus sign exponent indicates we are approaching three from the left, or using values slightly less than three. These correspond to the first part of the piecewise function, whose limit is 9 (substitution).

Which of the following is equal to ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?

Evaluate the limit:

$\lim_{x\rightarrow 3^-}f(x), f(x)=\left\{\begin{matrix} x^2, x<3\\ln(x) , x\geq 3 \end{matrix}\right.$

$f(x) = \left\{\begin{matrix} \frac{2}{x+2}&x<0 \ 0 &x= 0 \ 2x+1 & x > 0 \end{matrix}\right.$

Which of the following is equal to $\lim_{x\rightarrow 0 }f(x)$ ?