Chapter 2

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{x \rightarrow 2}(x+3) $$

Determine at which of the points \(-1,1,\) and 2 the given function \(f\) is continuous. $$ f(x)=\left\\{\begin{array}{cl} x^{2}-1 & \text { if } x<2 \\ 3 & \text { if } x=2 \\ 3 \sin (\pi / x) & \text { if } 2

D In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow 6} \frac{3}{(x-6)^{2}} $$

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 3} 2 x $$

In Exercises \(1-8\), simplify the given expression. $$ \sqrt{2}^{\sqrt{3}} \cdot \sqrt{2}^{\sqrt{3}} $$

In Exercises \(1-15,\) determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=n /\left(n^{2}+1\right) $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{s \rightarrow 9}\left(s^{2}-6 s+10\right) $$

Determine at which of the points \(-1,1,\) and 2 the given function \(f\) is continuous. $$ f(x)=\left\\{\begin{array}{cl} 1-x & \text { if } x \leq-1 \\ \sqrt{5-x^{2}} & \text { if }-1

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 1}(6 x-1) $$

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow-\infty} \frac{4 x}{4 x-7} $$

Simplify the given expression. $$ 4^{x} \cdot 4^{c} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=n+5 $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{h \rightarrow 4}\left(3 h^{2}+2 h+1\right) $$

Determine at which of the points \(-1,1,\) and 2 the given function \(f\) is continuous. $$ f(x)=\left\\{\begin{array}{cl} \left(x^{2}-1\right) /\left(x^{2}+1\right) & \text { if } x \leq-1 \\ (3 x+2)^{1 / 3} & \text { if }-1

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 6}(x / 3+2) $$

Simplify the given expression. $$ (1 / 8)^{-\pi / 3} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=3 n /(2 n+1) $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{s \rightarrow 0}(2+s) / s $$

Determine at which of the points \(-1,1,\) and 2 the given function \(f\) is continuous. $$ f(x)=\left\\{\begin{array}{cl} \left(x^{3}-x\right) /\left(x^{2}-1\right) & \text { if } x<-1 \\ 2 x+1 & \text { if }-1 \leq x \leq 2 \\ \left(x^{2}-4\right) /(x-2) & \text { if } 2

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow-1+}(x+1)^{-1} $$

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 12}(3-x / 4) $$

Simplify the given expression. $$ \left(8^{\sqrt{3}} \cdot 4^{\sqrt{7}}\right) / 2^{\pi} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=3-(-1)^{n} $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{h \rightarrow 1} \frac{h-3}{h+1} $$

Determine the values at which the given function \(f\) is continuous. Remember that if \(c\) is not in the domain of \(f,\) then \(f\) cannot be continuous at \(c .\) Also remember that the domain of a function that is defined by an expression consists of all real numbers at which the expression can be evaluated. $$ f(x)=x^{2}+4 $$

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow+\infty} \frac{x+\sqrt{x}}{x^{2}-\sqrt{x}} $$

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 0}(\sqrt{2}-\pi x) $$

Simplify the given expression. $$ \left(\sqrt{11}^{\sqrt{2}}\right)^{\sqrt{2}} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=1 /(n+2) $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{x \rightarrow-3} \frac{x-3}{x^{2}-9} $$

Determine the values at which the given function \(f\) is continuous. Remember that if \(c\) is not in the domain of \(f,\) then \(f\) cannot be continuous at \(c .\) Also remember that the domain of a function that is defined by an expression consists of all real numbers at which the expression can be evaluated. $$ f(x)=2 x-9 $$

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 3} \frac{x^{2}-4 x+3}{x-3} $$

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow \pi} \frac{3}{\sin (x)} $$

Simplify the given expression. $$ \left(\sqrt{e}^{\sqrt{2}}\right)^{2} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=1-1 / n $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \lim _{x \rightarrow 2} g(x) \text { for } $$ $$ g(x)=\left\\{\begin{aligned} 4 & \text { if } x<0 \\ -4 & \text { if } x \geq 0 \end{aligned}\right. $$

Determine the values at which the given function \(f\) is continuous. Remember that if \(c\) is not in the domain of \(f,\) then \(f\) cannot be continuous at \(c .\) Also remember that the domain of a function that is defined by an expression consists of all real numbers at which the expression can be evaluated. $$ f(x)=1 /(x+1) $$

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow \infty} \frac{x+\cos (x)}{x-\sin (x)} $$

Simplify the given expression. $$ \left(3^{4} \cdot 9^{3} / 27^{2}\right)^{1 / 2} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=\cos (n \pi) $$

Decide whether the indicated limit exists. If the limit does exist, compute it. $$ \begin{array}{l} \lim _{x \rightarrow-1} f(x) \text { for } \\ \qquad f(x)=\left\\{\begin{aligned} 6 & \text { if } x \leq-1 \\ 10 & \text { if } x>-1 \end{aligned}\right. \end{array} $$

Determine the values at which the given function \(f\) is continuous. Remember that if \(c\) is not in the domain of \(f,\) then \(f\) cannot be continuous at \(c .\) Also remember that the domain of a function that is defined by an expression consists of all real numbers at which the expression can be evaluated. $$ f(x)=x /\left(x^{2}-x-5\right) $$

In Exercises \(1-8\), evaluate the given limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2} $$

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow+\infty} \frac{\sqrt{x}}{x+1} $$

Simplify the given expression. $$ \left(\left(5^{3 / 4}\right)^{3} / \sqrt{\sqrt{5}}\right)^{1 / 2} $$

Determine whether the sequence \(\left\\{a_{n}\right\\}\) converges. If it does, state the limit. $$ a_{n}=\sin (n \pi) $$

Some algebraic manipulation is necessary to determine whether the indicated limit exists. If the limit does exist, compute it and supply reasons for each step of your answer. If the limit does not exist, explain why. $$ \lim _{x \rightarrow 5} \frac{\left(x^{2}-25\right)}{(x-5)} $$

Determine the values at which the given function \(f\) is continuous. Remember that if \(c\) is not in the domain of \(f,\) then \(f\) cannot be continuous at \(c .\) Also remember that the domain of a function that is defined by an expression consists of all real numbers at which the expression can be evaluated. $$ f(x)=1 /\left(x^{2}+1\right) $$

In Exercises \(1-20\), determine whether the given limit exists. If it does exist, then compute it. $$ \lim _{x \rightarrow 1^{+}} \frac{1}{\sqrt{x-1}} $$

In Exercises \(9-22,\) rewrite the given expression without using any exponentials or logarithms. $$ \log _{5}(1 / 125) $$