Use l’Hospital’s rule to find $\lim _{x \rightarrow \infty}\left(1+\frac{c}{x}\right)^{x}$ where c is a constant.

Find the indicated limit. Use l'Hospital's rule if necessary.

$$\lim _{x \rightarrow 0} \frac{x^2}{\cos 3 x-1}$$

Find the limit of the following. Use l'Hopital's rule if it applies.

$$\lim\limits_{x \rightarrow 0}\frac{e^{x}-1}{\sin x}$$

Use l'Hospital's rule where applicable to find each limit.

$$\lim _{x \rightarrow 0} \frac{x e^x}{e^x-1}$$

Use the General Power Rule to find the derivative of the function. $h(t)=\left(1-t^2\right)^4$

Use the given rule to write the 4th, 5th, 6th, and 7th terms of each sequence. $a_{n} = 2(n-1)^{3}$

Use L'Hospital's Rule to find the limit

$$\lim _{x \rightarrow 0^{+}} \frac{\ln \left(e^{x}-1\right)}{\ln x}$$

In given Problem use Leibniz’s rule to find $\frac {dy} {dx}$.
y = $\int_4^{x^2+1} \sqrt{t}$ dt

Use the product rule to differentiate

(a) $y=x^2(x+5)^3$

(b) $y=x^5(4 x+5)^2$

(c) $y=x \sqrt[4]{(x+1)}$

Let

$$g(x) = f^{-1}(x),$$

the inverse of function

$$f.$$

Write the rule for g for each function

$$f$$

given below.

$$f(x) = 5^x$$

$$f(x) = \log_4 x$$

$$f(x) = \log_e {x}$$

Use l'Hopital's Rule to find the limit. $\lim _{x \rightarrow \infty}\left(\sqrt{x^2+x+1}-\sqrt{x^2-x}\right)$

Let $f(x)=x^n$ and $g(x)=x^{1 / n}$. Compute $g^{\prime}(x)$ using the said theorem and check your answer using the Power Rule.

Sketch the graph of the function with the given rule. Find the domain and rang of the function.

$$f(x)=\left\{\begin{array}{ll} -x+1 & \text { if } x \leq 1 \\ x^{2}-1 & \text { if } x>1 \end{array}\right.$$

Explain why L'Hopital's Rule does not apply to $\lim _{x \rightarrow 0} \frac{x^{2} \sin \frac{1}{x}}{\sin x}$.