?A right circular cylinder with a height of 10 cm and a surface area of \(S

How is \(\lim \limits_{x \rightarrow a} f(x)\) calculated if f is a polynomial function?

How are \(\lim \limits_{x \rightarrow a^{-}} f(x)\) and \(\lim \limits_{x \rightarrow a^{+}} f(x)\) calculated if f is a polynomial function?

For what values of a does \(\lim \limits_{x \rightarrow a} r(x)=r(a)\) if r is a rational function?

Assume \(\lim \limits_{x \rightarrow 3} g(x)=4\) and f(x) = g(x) whenever \(x \neq 3\). Evaluate \(\lim \limits_{x \rightarrow 3} f(x)\), if possible.

Explain why \(\lim \limits_{x \rightarrow 3} \frac{x^{2}-7 x+12}{x-3}=\lim \limits_{x \rightarrow 3}(x-4)\).

If \(\lim \limits_{x \rightarrow 2} f(x)=-8\), find \(\lim \limits_{x \rightarrow 2}[f(x)]^{2 / 3}\).

Suppose p and q are polynomials. If \(\lim _{x \rightarrow 0} \frac{p(x)}{q(x)}=10\) and q(0) = 2, find p(0).

If \(\lim \limits_{x \rightarrow 2} f(x)=\lim \limits_{x \rightarrow 2} h(x)=5\), find \(\lim \limits_{x \rightarrow 2} g(x)\) when \(f(x) \leq g(x) \leq h(x)\) for all values of x.

Evaluate \(\lim \limits_{x \rightarrow 5} \sqrt{x^{2}-9}\).

Suppose \(f(x)=\left\{\begin{array}{ll} 4 & \text { if } x \leq 3 \\ x+2 & \text { if } x>3 \end{array}\right.\) Compute \(\lim \limits_{x \rightarrow 3^{-}} f(x)\) and \(\lim \limits_{x \rightarrow 3^{+}} f(x)\).

Evaluate the following limits. \(\lim \limits_{x \rightarrow 4}(3 x-7)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 1}(-2 x+5)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow-9}(5 x)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 2}(-3 x)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 6}(4)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow-5}(\pi)\)

Assume \(\lim \limits_{x \rightarrow 1} f(x)=8, \lim \limits_{x \rightarrow 1} g(x)=3\), and \(\lim \limits_{x \rightarrow 1} h(x)=2\). Compute the following limits and state the limit laws used to justify your computations. \(\lim \limits_{x \rightarrow 1}[4 f(x)]\)

Assume \(\lim \limits_{x \rightarrow 1} f(x)=8, \lim \limits_{x \rightarrow 1} g(x)=3\), and \(\lim \limits_{x \rightarrow 1} h(x)=2\). Compute the following limits and state the limit laws used to justify your computations. \(\lim \limits_{x \rightarrow 1}\left[\frac{f(x)}{h(x)}\right]\)

Assume \(\lim \limits_{x \rightarrow 1} f(x)=8, \lim \limits_{x \rightarrow 1} g(x)=3\), and \(\lim \limits_{x \rightarrow 1} h(x)=2\). Compute the following limits and state the limit laws used to justify your computations. \(\lim \limits_{x \rightarrow 1}\left[\frac{f(x) g(x)}{h(x)}\right]\)

Assume \(\lim \limits_{x \rightarrow 1} f(x)=8, \lim \limits_{x \rightarrow 1} g(x)=3\), and \(\lim \limits_{x \rightarrow 1} h(x)=2\). Compute the following limits and state the limit laws used to justify your computations. \(\lim \limits_{x \rightarrow 1}\left[\frac{f(x)}{g(x)-h(x)}\right]\)

Assume \(\lim \limits_{x \rightarrow 1} f(x)=8, \lim \limits_{x \rightarrow 1} g(x)=3\), and \(\lim \limits_{x \rightarrow 1} h(x)=2\). Compute the following limits and state the limit laws used to justify your computations. \(\lim \limits_{x \rightarrow 1}[h(x)]^{5}\)

Assume \(\lim \limits_{x \rightarrow 1} f(x)=8, \lim \limits_{x \rightarrow 1} g(x)=3\), and \(\lim \limits_{x \rightarrow 1} h(x)=2\). Compute the following limits and state the limit laws used to justify your computations. \(\lim \limits_{x \rightarrow 1} \sqrt[3]{f(x) g(x)+3}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 1}\left(2 x^{3}-3 x^{2}+4 x+5\right)\)

Evaluate the following limits. \(\lim \limits_{t \rightarrow-2}\left(t^{2}+5 t+7\right)\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 1} \frac{5 x^{2}+6 x+1}{8 x-4}\)

Evaluate the following limits. \(\lim \limits_{t \rightarrow 3} \sqrt[3]{t^{2}-10}\)

Evaluate the following limits. \(\lim \limits_{b \rightarrow 2} \frac{3 b}{\sqrt{4 b+1}-1}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 2}\left(x^{2}-x\right)^{5}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 3} \frac{-5 x}{\sqrt{4 x-3}}\)

Evaluate the following limits. \(\lim \limits_{h \rightarrow 0} \frac{3}{\sqrt{16+3 h}+4}\)

Let \(f(x)=\left\{\begin{array}{ll} x^{2}+1 & \text { if } x<-1 \\ \sqrt{x+1} & \text { if } x \geq-1 \end{array}\right.\) Compute the following limits or state that they do not exist. a. \(\lim \limits_{x \rightarrow-1^{-}} f(x)\) b. \(\lim \limits_{x \rightarrow-1^{+}} f(x)\) c. \(\lim \limits_{x \rightarrow-1} f(x)\)

Let \(f(x)=\left\{\begin{array}{ll} 0 & \text { if } x \leq-5 \\ \sqrt{25-x^{2}} & \text { if }-5<x<5 \\ 3 x & \text { if } x \geq 5 \end{array}\right.\) Compute the following limits or state that they do not exist. a. \(\lim \limits_{x \rightarrow-5^{-}} f(x)\) b. \(\lim \limits_{x \rightarrow-5^{+}} f(x)\) c. \(\lim \limits_{x \rightarrow-5} f(x)\) d. \(\lim \limits_{x \rightarrow 5^{-}} f(x)\) e. \(\lim \limits_{x \rightarrow 5^{+}} f(x)\) f. \(\lim \limits_{x \rightarrow 5} f(x)\)

a. Evaluate \(\lim \limits_{x \rightarrow 2^{+}} \sqrt{x-2}\). b. Why don't we consider evaluating \(\lim \limits_{x \rightarrow 2^{-}} \sqrt{x-2}\)?

a. Evaluate \(\lim \limits_{x \rightarrow 3^{-}} \sqrt{\frac{x-3}{2-x}}\). b. Why don't we consider evaluating \(\lim \limits_{x \rightarrow 3^{+}} \sqrt{\frac{x-3}{2-x}}\).

Show that \(\lim \limits_{x \rightarrow 0}|x|=0\) by first evaluating \(\lim \limits_{x \rightarrow 0^{-}}|x|\) and \(\lim \limits_{x \rightarrow 0^{+}}|x|\). Recall that \(|x|=\left\{\begin{array}{ll} -x & \text { if } x<0 \\ x & \text { if } x \geq 0 \end{array}\right.\).

Show that \(\lim \limits_{x \rightarrow a}|x|=|a|\) for any real number. (Hint: Consider three cases, a < 0, a = 0, and a > 0.)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow 1} \frac{x^{2}-1}{x-1}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow 3} \frac{x^{2}-2 x-3}{x-3}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow 4} \frac{x^{2}-16}{4-x}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{t \rightarrow 2} \frac{3 t^{2}-7 t+2}{2-t}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow b} \frac{(x-b)^{50}-x+b}{x-b}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow-b} \frac{(x+b)^{7}+(x+b)^{10}}{4(x+b)}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow-1} \frac{(2 x-1)^{2}-9}{x+1}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{h \rightarrow 0} \frac{\frac{1}{5+h}-\frac{1}{5}}{h}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{x \rightarrow 9} \frac{\sqrt{x}-3}{x-9}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{t \rightarrow a} \frac{\sqrt{3 t+1}-\sqrt{3 a+1}}{t-a}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim \limits_{h \rightarrow 0} \frac{\sqrt{16+h}-4}{h}\)

Evaluate the following limits, where a and b are fixed real numbers. \(\lim _{x \rightarrow 0} \frac{a-\sqrt{a^{2}-x^{2}}}{x^{2}}\)

a. Sketch a graph of \(y=2^{x}\) and carefully draw three secant lines connecting the points P(0,1) and \(Q\left(x, 2^{x}\right)\), for x = -3, -2, and -1. b. Find the slope of the line that joins P(0, 1) and \(Q\left(x, 2^{x}\right)\) for \(x \neq 0\). c. Complete the table and make a conjecture about the value of \(\lim \limits_{x \rightarrow 0^{-}} \frac{2^{x}-1}{x}\).

a. Sketch a graph of \(y=3^{x}\) and carefully draw four secant lines connecting the points P(0, 1) and \(Q\left(x, 3^{x}\right)\) for x = -2,-1,1, and 2. b. Find the slope of the line that joins P(0,1) and \(Q\left(x, 3^{x}\right)\) for \(x \neq 0\). c. Complete the table and make a conjecture about the value of \(\lim \limits_{x \rightarrow 0} \frac{3^{x}-1}{x}\).

a. Show that \(-|x| \leq x \sin \left(\frac{1}{x}\right) \leq|x|\) for all \(x \neq 0\). b. Illustrate the inequality \(-|x| \leq x \sin \left(\frac{1}{x}\right) \leq|x|\) with a graph. c. Use the Squeeze Theorem to show that \(\lim \limits_{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=0\).

It can be shown that \(1-x^{2} / 2 \leq \cos x \leq 1\) for x near 0. a. Illustrate the inequality \(1-x^{2} / 2 \leq \cos x \leq 1\) with a graph. b. Use the inequality in part (a) to find \(\lim \limits_{x \rightarrow 0} \cos x\).

It can be shown that \(1-\frac{x^{2}}{6} \leq \frac{\sin x}{x} \leq 1\) for x near 0. a. Illustrate the inequality \(1-\frac{x^{2}}{6} \leq \frac{\sin x}{x} \leq 1\) with a graph. b. Use the inequality in part (a) to find \(\lim \limits_{x \rightarrow 0} \frac{\sin x}{x}\).

a. Draw a graft to verify that \(-|x| \leq x^{2} \ln x^{2} \leq|x|\) for \(-1 \leq x \leq 1\). b. Use the Squeeze Theorem to determine \(\lim \limits_{x \rightarrow 0} x^{2} \ln x^{2}\).

Determine whether the following statements are true and give an explanation or counterexample. Assume a and L are finite numbers. a. If \(\lim \limits_{x \rightarrow a} f(x)=L\), then f(a) = L. b. If \(\lim \limits_{x \rightarrow a^{-}} f(x)=L\), then \(\lim \limits_{x \rightarrow a^{+}} f(x)=L\). c. If \(\lim \limits_{x \rightarrow a} f(x)=L\) and \(\lim \limits_{x \rightarrow a} g(x)=L\), then f(a) = g(a). d. The limit \(\lim \limits_{x \rightarrow a} \frac{f(x)}{g(x)}\) does not exist if g(a) = 0. e. If \(\lim \limits_{x \rightarrow 1^{+}} \sqrt{f(x)}=\sqrt{\lim \limits_{x \rightarrow 1^{+}} f(x)}\), it follows that \(\lim \limits_{x \rightarrow 1} \sqrt{f(x)}=\sqrt{\lim \limits_{x \rightarrow 1} f(x)}\).

Evaluate the following limits, where c and k are constants. \(\lim \limits_{h \rightarrow 0} \frac{100}{(10 h-1)^{11}+2}\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{x \rightarrow 2}(5 x-6)^{3 / 2}\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{x \rightarrow 5}(3 x-16)^{3 / 7}\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{x \rightarrow 1} \frac{\sqrt{10 x-9}-1}{x-1}\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{x \rightarrow 2}\left(\frac{1}{x-2}-\frac{2}{x^{2}-2 x}\right)\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{h \rightarrow 0} \frac{(5+h)^{2}-25}{h}\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{x \rightarrow c} \frac{x^{2}-2 c x+c^{2}}{x-c}\)

Evaluate the following limits, where c and k are constants. \(\lim \limits_{w \rightarrow-k} \frac{w^{2}+5 k w+4 k^{2}}{w^{2}+k w}\)

Suppose \(f(x)=\left\{\begin{array}{ll} 3 x+b & \text { if } x \leq 2 \\ x-2 & \text { if } x>2 \end{array}\right.\) Determine a value of the constant b for which \(\lim \limits_{x \rightarrow 2} f(x)\) exists and state the value of the limit, if possible.

Suppose \(g(x)=\left\{\begin{array}{ll} x^{2}-5 x & \text { if } x \leq-1 \\ a x^{3}-7 & \text { if } x>-1 \end{array}\right.\) Determine a value of the constant a for which \(\lim \limits_{x \rightarrow-1} g(x)\) exists and state the value of the limit, if possible.

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow 2} \frac{x^{5}-32}{x-2}\)

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow 1} \frac{x^{6}-1}{x-1}\)

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow-1} \frac{x^{7}+1}{x+1}\) (Hint: Use the formula for \(x^{7}-a^{7}\) with a = -1).

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow a} \frac{x^{5}-a^{5}}{x-a}\)

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}\), for any positive integer n

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1}\) (Hint: \(x-1=(\sqrt[3]{x})^{3}-(1)^{3}\))

Calculate the following limits using the factorization formula \(x^{n}-a^{n}=(x-a)\left(x^{n-1}+x^{n-2} a+x^{n-3} a^{2}+\cdots+x a^{n-2}+a^{n-1}\right)\), where n is a positive integer and a is a real number. \(\lim \limits_{x \rightarrow 16} \frac{\sqrt[4]{x}-2}{x-16}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 1} \frac{x-1}{\sqrt{x}-1}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 1} \frac{x-1}{\sqrt{4 x+5}-3}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 4} \frac{3(x-4) \sqrt{x+5}}{3-\sqrt{x+5}}\)

Evaluate the following limits. \(\lim \limits_{x \rightarrow 0} \frac{x}{\sqrt{c x+1}-1}\), where c is a constant

Give examples of functions f and g such that \(\lim \limits_{x \rightarrow 1} f(x)=0\) and \(\lim _{x \rightarrow 1}(f(x) \cdot g(x))=5\).

Give an example of a function f satisfying \(\lim \limits_{x \rightarrow 1}\left(\frac{f(x)}{x-1}\right)=2\).

Find constants b and c in the polynomial \(p(x)=x^{2}+b x+c\) such that \(\lim \limits_{x \rightarrow 2} \frac{p(x)}{x-2}=6\). Are the constants unique?

Suppose a spaceship of length \(L_{0}\) is traveling at a high rate of speed v relative to an observer. To the observer, the ship appears to have a smaller length given by the Lorentz contraction formula \(L=L_{0} \sqrt{1-\frac{v^{2}}{c^{2}}}\) where c is the speed of light. a. What is the observed length L of the ship if it is traveling at 50% of the speed of light? b. What is the observed length L of the ship if it is traveling at 75% of the speed of light? c. In parts (a) and (b), what happens to L as the speed of the ship increases? d. Find \(\lim \limits_{v \rightarrow c^{-}} L_{0} \sqrt{1-\frac{v^{2}}{c^{2}}}\) and explain the significance of this limit.

A right circular cylinder with a height of 10 cm and a surface area of \(S \mathrm{~cm}^{2}\) has a radius given by \(r(S)=\frac{1}{2}\left(\sqrt{100+\frac{2 S}{\pi}}-10\right)\). Find \(\lim \limits_{S \rightarrow 0^{+}} r(S)\) and interpret your result.

A cylindrical tank is filled with water to a depth of 9 m. At t = 0, a drain in the bottom of the tank is opened and water flows out of the tank. The depth of water in the tank (measured from the bottom of the tank) t seconds after the drain is opened is approximated by \(d(t)=(3-0.015 t)^{2} \quad \text { for } 0 \leq t \leq 200\). Evaluate and interpret \(\lim \limits_{t \rightarrow 200^{-}} d(t)\).

The magnitude of the electric field at a point x meters from the midpoint of a 0.1-m line of charge is given by \(E(x) = \frac{4.35}{x \sqrt{x^{2}+0.01}}\) (in units of newtons per coulomb, N/C). Evaluate \(\lim \limits_{x \rightarrow 10} E(x)\).

If \(\lim \limits_{x \rightarrow 1} f(x)=4\), find \(\lim \limits_{x \rightarrow-1} f\left(x^{2}\right)\).

Suppose g(x) = f(1 - x) for all x, \(\lim \limits_{x \rightarrow 1^{+}} f(x)=4\), and \(\lim \limits_{x \rightarrow 1^{-}} f(x)=6\). Find \(\lim \limits_{x \rightarrow 0^{+}} g(x) \text { and } \lim \limits_{x \rightarrow 0^{-}} g(x)\).

Consider the angle \(\theta\) in standard position in a unit circle where \(0 \leq \theta<\pi / 2 \text { or }-\pi / 2 \leq \theta<0\) (use both figures). a. Show that \(|A C|=|\sin \theta| \text { for }-\pi / 2<\theta<\pi / 2\). (Hint: Consider the cases \(0<\theta<\pi / 2 \text { and }-\pi / 2<\theta<0\) separately.) b. Show that \(|\sin \theta|<|\theta| \text { for }-\pi / 2<\theta<\pi / 2\). (Hint: The length of arc AB is \(\theta\) if \(0<\theta<\pi / 2\) and \(-\theta\) if \(-\pi / 2<\theta<0\).) c. Conclude that \(-|\theta| \leq \sin \theta \leq|\theta| \text { for }-\pi / 2<\theta<\pi / 2\). d. Show that \(0 \leq 1-\cos \theta \leq|\theta| \text { for }-\pi / 2<\theta<\pi / 2\).

Given the polynomial \(p(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}\) prove that \(\lim \limits_{x \rightarrow a} p(x)=p(a)\) for any value of a.