Limits (College Board AP® Calculus AB): Exam Questions

The graph of the function $f$ is shown above. Which of the following statements is false?

The line $y=3$ is a horizontal asymptote to the graph of which of the following functions?

If $f\left(x\right)=\{\begin{array}{l}\ln x\mathrm{for}0<x\le 3\\ x\ln 3\mathrm{for}3<x\le 5\end{array}$, then $\underset{x\to 3}{\lim }f\left(x\right)$ is

Let $f$ be a function such that $\underset{x\to 0}{\lim }f\left(x\right)=-1$.

Let $g$ be the function defined by $g\left(x\right)=\sin x$.

What is $\underset{x\to 0^{-}}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$?

Let $f$, $g$ and $h$ be functions defined on the interval $1<x<3$. $\underset{x\to 2}{\lim }g\left(x\right)=\underset{x\to 2}{\lim }h\left(x\right)=-5$, and it is also true that $g\left(x\right)\le f\left(x\right)\le h\left(x\right)$ for all $x$ such that $1<x<3$.

Which one of the following results allows you to determine that $\underset{x\to 2}{\lim }f\left(x\right)=-5$?

$\underset{x\to \infty }{\lim }\frac{\left(3x-2\right)\left(1-x\right)}{\left(x-2\right)\left(x+1\right)}$ is

$\underset{x\to 0}{\lim }\frac{7x^5-3x^4}{9x^5+2x^4}$ is

The domain of a function $f$ is $x\le 0$. The horizontal line $y=-1$ is an asymptote for the graph of $f$. Which of the following statements must be true?

For which of the following does $\underset{x\to 2}{\lim }f\left(x\right)$ exist?

If $\underset{x\to 2}{\lim }f\left(x\right)=5$, which of the following must be true?

I. $f$ is continuous at $x=2$.

II. $f$ is differentiable at $x=2$.

III. $f\left(2\right)=5$

Let $f$, $g$ and $h$ be continuous functions such that $\underset{x\to 5}{\lim }f\left(x\right)=-2$, $\underset{x\to 3}{\lim }g\left(x\right)=5$ and $\underset{x\to 3}{\lim }h\left(x\right)=-7$.

If the function $p$ is defined by $p\left(x\right)=\frac{f\left(g\left(x\right)\right)}{{\left[h\left(x\right)\right]}^2}$, then $\underset{x\to 3}{\lim }p\left(x\right)$ is

$\underset{x\to 2}{\lim }\left(\frac{\sqrt{x+1}-\sqrt{3}}{x-2}\right)=$

Let $f$ and $g$ be the functions defined by $f\left(x\right)=x^2-2x+2$ and $g\left(x\right)=\sin \left(\frac{\pi x}{2}\right)$. Let $h$ be a function defined for all real numbers and satisfying $f\left(x\right)\le h\left(x\right)\le g\left(x\right)$ for all values of $x$ except $x=1$.

What is $\underset{x\to 1}{\lim }h\left(x\right)$?

If

$f\left(x\right)=\{\begin{array}{l}\frac{\sin \left(7-7x\right)}{\sin \left(2x-2\right)}\mathrm{for}-\frac{1}{2}\le x<1\\ 0\mathrm{for}x=1\\ \frac{x^2-3x+9}{2}\mathrm{for}x>1\end{array}$

then $\underset{x\to 1}{\lim }f\left(x\right)$ is

The graph of a function $f$ is shown above. For which of the following values of $c$ does $\underset{x\to c}{\lim }f\left(x\right)=2$?

What are all the horizontal asymptotes of the graph of $y=\frac{3+5^x}{2+5^x}$ in the $xy$-plane.