L'Hospital's Rule (College Board AP® Calculus AB): Free Response Questions

Find the value of $\underset{x\to 0}{\lim }\left(\frac{\sin x}{x^2+2x}\right)$, or show that it does not exist. Justify your answer.

Use L'Hospital's rule to find the value of $\underset{x\to \infty }{\lim }\left(\frac{7x-3x^2}{12x^2+5x}\right)$.

Find the value of $\underset{x\to 0}{\lim }\left(\frac{-\sin \left(5\pi x\right)}{2\pi x}\right)$, or show that it does not exist. Justify your answer.

Functions $f$ and $g$ are differentiable functions with $g\left(2\right)=1.$

The function $g$ satisfies $g\left(x\right)=\frac{4-x^2}{f\left(x\right)-3}$ for $x\ne 2$. It is known that $\underset{x\to 2}{\lim }g\left(x\right)$ can be evaluated using L'Hospital's rule.

Use $\underset{x\to 2}{\lim }g\left(x\right)$ to find $f\left(2\right)$, showing the work that leads to your answer.

Find the value of $\underset{x\to 0}{\lim }\left(\frac{2e^x\left(e^x-1\right)}{3\sin x{\cos }^{2}x}\right)$ or show that it does not exist. Justify your answer.

The function $f$ is defined on the closed interval $\left[0,9\right]$ and satisfies $f\left(3\right)=2$. The graph of $f^{\text{'}}$, the derivative of $f$, consists of two line segments and a semicircle, as shown in the figure above.

Find the value of $\underset{x\to 3}{\lim }\left(\frac{2x-3f\left(x\right)}{x^2-x-6}\right)$, or show that it does not exist. Justify your answer.

Let $f$ be a continuous function defined on the closed interval $-2\le x\le 8$. The graph of $f$, consisting of three line segments, is shown in the figure above. Let $G$ be the function defined by $G\left(x\right)={\int }_0^{x}f\left(t\right)dt$.

Find $\underset{x\to 4}{\lim }\left(\frac{x^2-3x-4}{G\left(x\right)}\right)$.

Functions $f$ and $g$ are differentiable functions with $g\left(3\right)=1.$

The function $g$ satisfies $g\left(x\right)=\frac{9-x^2}{{\left(f\left(x\right)\right)}^3-8}$ for $x\ne 3$. It is known that $\underset{x\to 3}{\lim }g\left(x\right)$ can be evaluated using L'Hospital's rule. Use $\underset{x\to 3}{\lim }g\left(x\right)$ to find $f\left(3\right)$ and $f^{\text{'}}\left(3\right)$. Show the work that leads to your answers.

Let $f$ be the function defined by $f\left(x\right)=e^x\sin \left(2x\right)$.

Let $g$ be a differentiable function such that $g\left(\pi \right)=0$. The function $g\text{'}$, the derivative of $g$, is a piecewise linear function such that:

• $g\text{'}\left(x\right)=x-2\mathrm{on}\left(0,\pi \right]$

• $g\text{'}\left(x\right)=-\frac{x}{2}+\frac{3\pi }{2}-2\mathrm{on}\left(\pi ,2\pi \right]$

Find the value of $\underset{x\to \pi }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ or state that it does not exist. Justify your answer.

Evaluate $\underset{x\to \infty }{\lim }{\left[e^x+2x\right]}^{\frac{1}{x}}$ using L'Hospital's Rule.