Chapter 2

In Problems 1-10, simplify the given expression. \(10^{2 \log _{10} 5}\)

In Problems 1-6, find the indicated limit. $$ \lim _{x \rightarrow 3}(x-5) $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ f(x)=(x-3)(x-4) $$

Evaluate each limit. $$ \lim _{x \rightarrow 0} \frac{\cos x}{x+1} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x}{x-5}\)

In Problems 1-10, simplify the given expression. \(2^{2 \log _{2} x}\)

In Problems 1-6, find the indicated limit. $$ \lim _{t \rightarrow-1}(1-2 t) $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ g(x)=x^{2}-9 $$

Evaluate each limit. $$ \lim _{\theta \rightarrow \pi / 2} \theta \cos \theta $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x^{2}}{5-x^{3}}\)

In Problems 1-10, simplify the given expression. \(e^{3 \ln x}\)

In Problems 1-6, find the indicated limit. $$ \lim _{x \rightarrow-2}\left(x^{2}+2 x-1\right) $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ h(x)=\frac{3}{x-3} $$

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\cos ^{2} t}{1+\sin t} $$

Find the limits. \(\lim _{t \rightarrow-\infty} \frac{t^{2}}{7-t^{2}}\)

In Problems 1-10, simplify the given expression. \(e^{-2 \ln x}\)

Evaluate each limit. $$ \lim _{x \rightarrow 0} \frac{3 x \tan x}{\sin x} $$

Find the limits. \(\lim _{t \rightarrow-\infty} \frac{t}{t-5}\)

In Problems 1-10, simplify the given expression. \(\ln e^{\cos x}\)

In Problems 1-6, find the indicated limit. $$ \lim _{t \rightarrow-1}\left(t^{2}-1\right) $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ h(t)=\frac{|t-3|}{t-3} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x^{2}}{(x-5)(3-x)}\)

In Problems 1-10, simplify the given expression. \(\ln e^{-2 x-3}\)

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ h(t)=\frac{\left|\sqrt{(t-3)^{4}}\right|}{t-3} $$

Evaluate each limit. $$ \lim _{\theta \rightarrow 0} \frac{\sin 3 \theta}{2 \theta} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x^{2}}{x^{2}-8 x+15}\)

In Problems 1-10, simplify the given expression. \(\ln \left(x^{3} e^{-3 x}\right)\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2} $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ f(t)=|t| $$

Evaluate each limit. $$ \lim _{\theta \rightarrow 0} \frac{\sin 3 \theta}{\tan \theta} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{x^{3}}{2 x^{3}-100 x^{2}}\)

In Problems 1-10, simplify the given expression. \(e^{x-\ln x}\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{t \rightarrow-7} \frac{t^{2}+4 t-21}{t+7} $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ g(t)=|t-2| $$

Evaluate each limit. $$ \lim _{\theta \rightarrow 0} \frac{\tan 5 \theta}{\sin 2 \theta} $$

Find the limits. \(\lim _{\theta \rightarrow-\infty} \frac{\pi \theta^{5}}{\theta^{5}-5 \theta^{4}}\)

In Problems 1-10, simplify the given expression. \(e^{\ln 3+2 \ln x}\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow-1} \frac{x^{3}-4 x^{2}+x+6}{x+1} $$

Evaluate each limit. $$ \lim _{\theta \rightarrow 0} \frac{\cot (\pi \theta) \sin \theta}{2 \sec \theta} $$

Find the limits. \(\lim _{x \rightarrow \infty} \frac{3 x^{3}-x^{2}}{\pi x^{3}-5 x^{2}}\)

In Problems 1-10, simplify the given expression. \(e^{\ln x^{2}-y \ln x}\)

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow 0} \frac{x^{4}+2 x^{3}-x^{2}}{x^{2}} $$

$$ f(x)=\frac{0}{x} $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ f(x)=\frac{21-7 x}{x-3} $$

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\sin ^{2} 3 t}{2 t} $$

Find the limits. \(\lim _{\theta \rightarrow \infty} \frac{\sin ^{2} \theta}{\theta^{2}-5}\)

$$ \lim _{x \rightarrow 0}(2 x-1)=-1 $$

In Problems 1-15, state whether the indicated function is continu ous at 3. If it is not continuous, tell why. $$ r(t)= \begin{cases}\frac{t^{3}-27}{t-3} & \text { if } t \neq 3 \\ 27 & \text { if } t=3\end{cases} $$

Evaluate each limit. $$ \lim _{t \rightarrow 0} \frac{\tan ^{2} 3 t}{2 t} $$

In Problems 7-18, find the indicated limit. In most cases, it will be wise to do some algebra first (see Example 2). $$ \lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3} $$