Chapter 1

Evaluate the given limit. $$ \lim _{x \rightarrow 3} 4^{x^{3}-8 x} $$

A function \(f\) and \(a\) value \(a\) are given. Approximate the limit of the difference quotient, \(\lim _{h \rightarrow 0} \frac{f(a+h)-f(a)}{h},\) using \(h=\pm 0.1, \pm 0.01 .\) $$ f(x)=\cos x, \quad a=\pi $$

Evaluate the given limit. $$ \lim _{x \rightarrow \infty} \frac{x^{3}+2 x^{2}+1}{x-5} $$

Give the intervals on which the given function is continuous. $$ g(x)=\sqrt{4-x^{2}} $$

Approximate the limit numerically: \(\lim _{x \rightarrow 0.4} \frac{x^{2}-4.4 x+1.6}{x^{2}-0.4 x}\).

Evaluate the given limit. $$ \lim _{x \rightarrow \pi / 6} \csc x $$

Evaluate the given limit. $$ \lim _{x \rightarrow \infty} \frac{x^{3}+2 x^{2}+1}{5-x} $$

Give the intervals on which the given function is continuous. $$ h(k)=\sqrt{1-k}+\sqrt{k+1} $$

Approximate the limit numerically: \(\lim _{x \rightarrow 0.2} \frac{x^{2}+5.8 x-1.2}{x^{2}-4.2 x+0.8}\).

Evaluate the given limit. $$ \lim _{x \rightarrow 0} \ln (1+x) $$

Evaluate the given limit. $$ \lim _{x \rightarrow-\infty} \frac{x^{3}+2 x^{2}+1}{x^{2}-5} $$

Give the intervals on which the given function is continuous. $$ f(t)=\sqrt{5 t^{2}-30} $$

Evaluate the given limit. $$ \lim _{x \rightarrow \pi} \frac{x^{2}+3 x+5}{5 x^{2}-2 x-3} $$

Evaluate the given limit. $$ \lim _{x \rightarrow-\infty} \frac{x^{3}+2 x^{2}+1}{5-x^{2}} $$

Give the intervals on which the given function is continuous. $$ g(t)=\frac{1}{\sqrt{1-t^{2}}} $$

Evaluate the given limit. $$ \lim _{x \rightarrow \pi} \frac{3 x+1}{1-x} $$

Use an \(\varepsilon-\delta\) proof to show that \(\lim _{x \rightarrow 1} 5 x-2=3\)

Give the intervals on which the given function is continuous. $$ g(x)=\frac{1}{1+x^{2}} $$

Evaluate the given limit. $$ \lim _{x \rightarrow 6} \frac{x^{2}-4 x-12}{x^{2}-13 x+42} $$

Give the intervals on which the given function is continuous. $$ f(x)=e^{x} $$

Evaluate the given limit. $$ \lim _{x \rightarrow 0} \frac{x^{2}+2 x}{x^{2}-2 x} $$

Let \(f(x)=\left\\{\begin{array}{ll}x^{2}-1 & x<3 \\ x+5 & x \geq 3\end{array}\right.\) Is \(f\) continuous everywhere?

Give the intervals on which the given function is continuous. $$ g(s)=\ln s $$

Evaluate the given limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}+6 x-16}{x^{2}-3 x+2} $$

Evaluate the limit: \(\lim _{x \rightarrow e} \ln x\).

Give the intervals on which the given function is continuous. $$ h(t)=\cos t $$

Evaluate the given limit. $$ \lim _{x \rightarrow 2} \frac{x^{2}-10 x+16}{x^{2}-x-2} $$

Give the intervals on which the given function is continuous. $$ f(k)=\sqrt{1-e^{k}} $$

Evaluate the given limit. $$ \lim _{x \rightarrow-2} \frac{x^{2}-5 x-14}{x^{2}+10 x+16} $$

Give the intervals on which the given function is continuous. $$ f(x)=\sin \left(e^{x}+x^{2}\right) $$

Evaluate the given limit. $$ \lim _{x \rightarrow-1} \frac{x^{2}+9 x+8}{x^{2}-6 x-7} $$

Test your understanding of the Intermediate Value Theorem. Let \(f\) be continuous on [1,5] where \(f(1)=-2\) and \(f(5)=\) -10. Does a value \(1

Use the Squeeze Theorem, where appropriate, to evaluate the given limit. $$ \lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right) $$

Test your understanding of the Intermediate Value Theorem. Let \(g\) be continuous on [-3,7] where \(g(0)=0\) and \(g(2)=\) 25. Does a value \(-3

Use the Squeeze Theorem, where appropriate, to evaluate the given limit. $$ \lim _{x \rightarrow 0} \sin x \cos \left(\frac{1}{x^{2}}\right) $$

Test your understanding of the Intermediate Value Theorem. Let \(f\) be continuous on [-1,1] where \(f(-1)=-10\) and \(f(1)=10\). Does a value \(-1

Use the Squeeze Theorem, where appropriate, to evaluate the given limit. $$ \lim _{x \rightarrow 1} f(x), \text { where } 3 x-2 \leq f(x) \leq x^{3} $$

Use the Squeeze Theorem, where appropriate, to evaluate the given limit. $$ \lim _{x \rightarrow 3} f(x), \text { where } 6 x-9 \leq f(x) \leq x^{2} $$

Use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval. $$ f(x)=x^{2}+2 x-4 \text { on }[1,1.5] $$

Challenge your understanding of limits but can be evaluated using the knowledge gained in this section. $$ \lim _{x \rightarrow 0} \frac{\sin 3 x}{x} $$

Use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval. $$ f(x)=\sin x-1 / 2 \text { on }[0.5,0.55] $$

Challenge your understanding of limits but can be evaluated using the knowledge gained in this section. $$ \lim _{x \rightarrow 0} \frac{\sin 5 x}{8 x} $$

Use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval. $$ f(x)=e^{x}-2 \text { on }[0.65,0.7] $$

Challenge your understanding of limits but can be evaluated using the knowledge gained in this section. $$ \lim _{x \rightarrow 0} \frac{\ln (1+x)}{x} $$

Use the Bisection Method to approximate, accurate to two decimal places, the value of the root of the given function in the given interval. $$ f(x)=\cos x-\sin x \text { on }[0.7,0.8] $$

Let \(f(x)=\left\\{\begin{array}{cc}x^{2}-5 & x<5 \\ 5 x & x \geq 5\end{array}\right.\). (a) \(\lim _{x \rightarrow 5^{-}} f(x)\) (c) \(\lim _{x \rightarrow 5} f(x)\) (b) \(\lim _{x \rightarrow 5^{+}} f(x)\) (d) \(f(5)\)

Challenge your understanding of limits but can be evaluated using the knowledge gained in this section. Let \(f(x)=0\) and \(g(x)=\frac{x}{x}\) (a) Show why \(\lim _{x \rightarrow 2} f(x)=0\). (b) Show why \(\lim _{x \rightarrow 0} g(x)=1\). (c) Show why \(\lim _{x \rightarrow 2} g(f(x))\) does not exist. (d) Show why the answer to part (c) does not violate the Composition Rule of Theorem 1.3.1.

Numerically approximate the following limits: (a) \(\lim _{x \rightarrow-4 / 5^{+}} \frac{x^{2}-8.2 x-7.2}{x^{2}+5.8 x+4}\) (b) \(\lim _{x \rightarrow-4 / 5^{-}} \frac{x^{2}-8.2 x-7.2}{x^{2}+5.8 x+4}\)

Give an example of function \(f(x)\) for which \(\lim _{x \rightarrow 0} f(x)\) does not exist.