Chapter 2

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

Give the appropriate \(\varepsilon-\delta\) definition of each statement. $$ \lim _{t \rightarrow a} f(t)=M $$

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

In Problems \(1-15\), state whether the indicated function is continuous at \(3 .\) If it is not contimuous, tell why. $$ f(x)=(x-3)(x-4) \quad \text { 2. } g(x)=x^{2}-9 $$

Simplify the given expression. $$ 10^{2 \log _{10} 5} $$

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

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ g(x)=x^{2}-9 $$

Simplify the given expression. $$ 2^{2} \log _{2} x $$

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

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ h(x)=\frac{3}{x-3} $$

Simplify the given expression. $$ e^{3 \ln x} $$

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ g(t)=\sqrt{t-4} $$

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

Give the appropriate \(\varepsilon-\delta\) definition of each statement. $$ \lim _{x \rightarrow c^{-}} f(x)=L $$

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ h(t)=\frac{|t-3|}{t-3} $$

Simplify the given expression. $$ \ln e^{\cos x} $$

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ h(t)=\frac{\mid \sqrt{(t-3)^{4} \mid}}{t-3} $$

Simplify the given expression. $$ \ln e^{-2 x-3} $$

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

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

Plot the function \(f(x)\) over the interval \([1.5,2.5] .\) Zoom in on the graph of each function to determine how close \(x\) must be to 2 in order that \(f(x)\) is within \(0.002\) of 4 . $$ f(x)=2 x $$

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(t)=|t| $$

Simplify the given expression. $$ \ln \left(x^{3} e^{-3 x}\right) $$

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ g(t)=|t-2| $$

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

Simplify the given expression. $$ e^{x-\ln x} $$

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ h(x)=\frac{x^{2}-9}{x-3} $$

Plot the function \(f(x)\) over the interval \([1.5,2.5] .\) Zoom in on the graph of each function to determine how close \(x\) must be to 2 in order that \(f(x)\) is within \(0.002\) of 4 . $$ f(x)=\sqrt{8 x} $$

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

Simplify the given expression. $$ e^{\ln 3+2 \ln x} $$

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

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

State whether the indicated function is continuous at \(3 .\) If it is not continuous, tell why. $$ f(x)=\frac{21-7 x}{x-3} $$

Plot the function \(f(x)\) over the interval \([1.5,2.5] .\) Zoom in on the graph of each function to determine how close \(x\) must be to 2 in order that \(f(x)\) is within \(0.002\) of 4 . $$ f(x)=\frac{8}{x} $$

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