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

Particle $P$ moves along the $x$-axis such that, for time $t>0$, its position is given by $x_P\left(t\right)=2+3e^{-t}$.

Particle $Q$ moves along the $y$-axis such that, for time $t>0$, its position is given by $y_Q\left(t\right)=\frac{1}{t}-3$.

As $t\to \infty$, which particle will eventually be farther from the origin? Give a reason for your answer.

The continuous function $f$ is defined on the closed interval $-2\le x\le 6$. The figure above shows the graph of $f$, consisting of two line segments and a quarter of a circle centered at the point $\left(6,-3\right)$.

Find $\underset{x\to 1}{\lim }\left(\frac{5^x-2f\left(x\right)}{\arctan x+3f^{\text{'}}\left(x\right)}\right)$.

Let $f$ and $g$ be the functions defined by $f\left(x\right)=3+\cos x$ and $g\left(x\right)=2-{\left(x-\pi \right)}^2$. It is known that $g\left(x\right)\le f\left(x\right)$ for $0<x<2\pi$.

Let $h$ be a function such that $g\left(x\right)\le h\left(x\right)\le f\left(x\right)$ for $0<x<2\pi$.

Find $\underset{x\to \pi }{\lim }h\left(x\right)$, being sure to justify your answer.

Find $\underset{x\to \infty }{\lim }\frac{2x^{15}-3}{5x^{15}+7}$ or show that it does not exist.

Find $\underset{x\to 3}{\lim }\frac{2x^2-x-15}{x^2-6x+9}$ or show that it does not exist.

Find $\underset{x\to 0}{\lim }\left(\frac{3x}{\sqrt{2}-\sqrt{2-x}}\right)$.

It is known that $\cos x\le \frac{\sin x}{x}\le \frac{1}{\cos x}$ for $-\frac{\pi }{2}<x<0$ and $0<x<\frac{\pi }{2}$.

Find $\underset{x\to 0}{\lim }\frac{\sin x}{x}$. Justify your answer.

Find $\underset{x\to 0}{\lim }\frac{\sin x\cos 2x}{3x}$.

The continuous function $f$ is defined on the closed interval $-4\le x\le 4$. The figure above shows the graph of $f$, consisting of two line segments which intersect at $\left(0,0\right)$. Let $g$ be the function defined by $g\left(x\right)=f^{\text{'}}\left(x\right)$.

Find $\underset{x\to 0}{\lim }g\left(x\right)$ or show that it does not exist.

Find $\underset{x\to \pi }{\lim }\frac{\sqrt{e^{\sin x}+2}-\sqrt{3}}{e^{\sin x}\cos x-\cos x}$.

Find $\underset{x\to \infty }{\lim }\left(\sqrt{4x^2+5x}-2x\right)$ or show that it does not exist.