💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!
Section 6
Limits Involving Infinity
For the function $f$ whose graph is given, state the following.(a) $$\lim _{x \rightarrow \infty} f(x)$$(b) $$\lim _{x \rightarrow-\infty} f(x)$$(c) $$\lim _{x \rightarrow 1} f(x)$$(d) $$\lim _{x \rightarrow 3} f(x)$$(e) The equations of the asymptotes
For the function $g$ whose graph is given, state the following.(a) $$\lim _{x \rightarrow \infty} g(x)$$(b) $$\lim _{x \rightarrow-\infty} g(x)$$(c) $$\lim _{x \rightarrow 0} g(x)$$(d) $$\lim _{x \rightarrow 2^{-}} g(x)$$(e) $$\lim _{x \rightarrow 2^{+}} g(x)$$(f) The equations of the asymptotes
Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.$$\lim _{x \rightarrow 0} f(x)=-\infty, \quad \lim _{x \rightarrow-\infty} f(x)=5, \quad \lim _{x \rightarrow \infty} f(x)=-5$$
Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.$$\lim _{x \rightarrow 2} f(x)=\infty, \quad \lim _{x \rightarrow-2^{+}} f(x)=\infty, \quad \lim _{x \rightarrow-2^{-}} f(x)=-\infty,$$ $$\lim _{x \rightarrow-\infty} f(x)=0, \quad \lim _{x \rightarrow \infty} f(x)=0, \quad f(0)=0$$
Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.$$\lim _{x \rightarrow 2} f(x)=-\infty, \quad \lim _{x \rightarrow \infty} f(x)=\infty, \quad \lim _{x \rightarrow-\infty} f(x)=0$$ $$\lim _{x \rightarrow 0^{+}} f(x)=\infty, \lim _{x \rightarrow 0^{-}} f(x)=-\infty$$
Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.$$\lim _{x \rightarrow \infty} f(x)=3, \quad \lim _{x \rightarrow 2^{-}} f(x)=\infty, \quad \lim _{x \rightarrow 2^{+}} f(x)=-\infty, \quad f$$ is odd
Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.$$f(0)=3, \quad \lim _{x \rightarrow 0^{-}} f(x)=4, \quad \lim _{x \rightarrow 0^{+}} f(x)=2$$ $$\lim _{x \rightarrow-\infty} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{+}} f(x)=\infty$$ $$\lim _{x \rightarrow \infty} f(x)=3$$
Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.$$\lim _{x \rightarrow 3} f(x)=-\infty, \quad \lim _{x \rightarrow \infty} f(x)=2, \quad f(0)=0, \quad f$$ is even
Guess the value of the limit$$\lim _{x \rightarrow \infty} \frac{x^{2}}{2^{x}}$$by evaluating the function $f(x)=x^{2} / 2^{x}$ for $x=0,1,2,3,$ $4,5,6,7,8,9,10,20,50,$ and $100 .$ Then use a graph of $f$ to support your guess.
Determine $\lim _{x \rightarrow 1^{-}} \frac{1}{x^{3}-1}$ and $\lim _{x \rightarrow 1^{+}} \frac{1}{x^{3}-1}$(a) by evaluating $f(x)=1 /\left(x^{3}-1\right)$ for values of $x$ that approach 1 from the left and from the right,(b) by reasoning as in Example $1,$ and(c) from a graph of $f .$
Use a graph to estimate all the vertical and horizontal asymptotes of the curve$$y=\frac{x^{3}}{x^{3}-2 x+1}$$
(a) Use a graph of$$f(x)=\left(1-\frac{2}{x}\right)^{x}$$to estimate the value of $\lim _{x \rightarrow \infty} f(x)$ correct to two decimal places.(b) Use a table of values of $f(x)$ to estimate the limit to four decimal places.
Find the limit.$$\lim _{x \rightarrow-3^{+}} \frac{x+2}{x+3}$$
Find the limit.$$\lim _{x \rightarrow-3^{-}} \frac{x+2}{x+3}$$
Find the limit.$$\lim _{x \rightarrow 1} \frac{2-x}{(x-1)^{2}}$$
Find the limit.$$\lim _{x \rightarrow \pi^{-}} \cot x$$
Find the limit.$$\lim _{x \rightarrow 2 \pi^{-}} x \csc x$$
Find the limit.$$\lim _{x \rightarrow 2^{-}} \frac{x^{2}-2 x}{x^{2}-4 x+4}$$
Find the limit.$$\lim _{x \rightarrow \infty} \frac{3 x-2}{2 x+1}$$
Find the limit.$$\lim _{x \rightarrow \infty} \frac{1-x^{2}}{x^{3}-x+1}$$
Find the limit.$$\lim _{t \rightarrow \infty} \frac{\sqrt{t}+t^{2}}{2 t-t^{2}}$$
Find the limit.$$\lim _{t \rightarrow \infty} \frac{t-t \sqrt{t}}{2 t^{3 / 2}+3 t-5}$$
Find the limit.$$\lim _{x \rightarrow \infty} \frac{\left(2 x^{2}+1\right)^{2}}{(x-1)^{2}\left(x^{2}+x\right)}$$
Find the limit.$$\lim _{x \rightarrow \infty} \frac{x^{2}}{\sqrt{x^{4}+1}}$$
Find the limit.$$\lim _{x \rightarrow \infty}\left(\sqrt{9 x^{2}+x}-3 x\right)$$
Find the limit.$$\lim _{x \rightarrow \infty}\left(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}\right)$$
Find the limit.$$\lim _{x \rightarrow \infty} \frac{x^{4}-3 x^{2}+x}{x^{3}-x+2}$$
Find the limit.$$\lim _{x \rightarrow \infty} \frac{\sin ^{2} x}{x^{2}}$$
Find the limit.$$\lim _{x \rightarrow \infty} \cos x$$
Find the limit.$$\lim _{x \rightarrow-\infty} \frac{1+x^{6}}{x^{4}+1}$$
Find the limit.$$\lim _{x \rightarrow \infty}(x-\sqrt{x})$$
Find the limit.$$\lim _{x \rightarrow \infty}\left(x^{2}-x^{4}\right)$$
Find the limit.$$\lim _{x \rightarrow-\infty}\left(x^{4}+x^{5}\right)$$
(a) Graph the function$$f(x)=\frac{\sqrt{2 x^{2}+1}}{3 x-5}$$How many horizontal and vertical asymptotes do you observe? Use the graph to estimate the values of the limits$$\lim _{x \rightarrow \infty} \frac{\sqrt{2 x^{2}+1}}{3 x-5} \quad \text { and } \quad \lim _{x \rightarrow-\infty} \frac{\sqrt{2 x^{2}+1}}{3 x-5}$$(b) By calculating values of $f(x),$ give numerical estimates of the limits in part (a).(c) Calculate the exact values of the limits in part (a). Did you get the same value or different values for these two limits? [In view of your answer to part (a), you might have to check your calculation for the second limit.
Find the horizontal and vertical asymptotes of each curve. Check your work by graphing the curve and estimating the asymptotes.$y=\frac{2 x^{2}+x-1}{x^{2}+x-2}$
Find the horizontal and vertical asymptotes of each curve. Check your work by graphing the curve and estimating the asymptotes.$F(x)=\frac{x-9}{\sqrt{4 x^{2}+3 x+2}}$
(a) Estimate the value of$$\lim _{x \rightarrow-\infty}\left(\sqrt{x^{2}+x+1}+x\right)$$by graphing the function $f(x)=\sqrt{x^{2}+x+1}+x.$(b) Use a table of values of $f(x)$ to guess the value of the limit.(c) Prove that your guess is correct.
(a) Use a graph of$$f(x)=\sqrt{3 x^{2}+8 x+6}-\sqrt{3 x^{2}+3 x+1}$$to estimate the value of $\lim _{x \rightarrow \infty} f(x)$ to one decimal place.(b) Use a table of values of $f(x)$ to estimate the limit to four decimal places.(c) Find the exact value of the limit.
Estimate the horizontal asymptote of the function$$f(x)=\frac{3 x^{3}+500 x^{2}}{x^{3}+500 x^{2}+100 x+2000}$$by graphing $f$ for $-10 \leqslant x \leqslant 10 .$ Then calculate the equation of the asymptote by evaluating the limit. How do you explain the discrepancy?
Find a formula for a function that has vertical asymptotes $x=1$ and $x=3$ and horizontal asymptote $y=1.$
Find a formula for a function $f$ that satisfies the following conditions:$$\begin{array}{ll}{\lim _{x \rightarrow \pm \infty} f(x)=0,} & {\lim _{x \rightarrow 0} f(x)=-\infty, \quad f(2)=0} \\ {\lim _{x \rightarrow 3^{-}} f(x)=\infty,} & {\lim _{x \rightarrow 3^{+}} f(x)=-\infty}\end{array}$$
Evaluate the limits.$$ \text { (a) }\lim _{x \rightarrow \infty} x \sin \frac{1}{x} \quad \text { (b) } \lim _{x \rightarrow \infty} \sqrt{x} \sin \frac{1}{x}$$
A function $f$ is a ratio of quadratic functions and has a vertical asymptote $x=4$ and just one $x$ -intercept, $x=1 .$ It is known that $f$ has a removable discontinuity at $x=-1$ and $\lim _{x \rightarrow-1} f(x)=2 .$ Evaluate$$\text { (a) } f(0) \quad \text { (b) } \lim _{x \rightarrow \infty} f(x)$$
By the end behavior of a function we mean the behavior of its values as $x \rightarrow \infty$ and as $x \rightarrow-\infty$ .(a) Describe and compare the end behavior of the functions$$P(x)=3 x^{5}-5 x^{3}+2 x \quad Q(x)=3 x^{5}$$by graphing both functions in the viewing rectangles $[-2,2]$ by $[-2,2]$ and $[-10,10]$ by $[-10,000,10,000]$(b) Two functions are said to have the same end behavior if their ratio approaches 1 as $x \rightarrow \infty .$ Show that $P$ and $Q$ have the same end behavior.
Let $P$ and $Q$ be polynomials. Find$$\lim _{x \rightarrow \infty} \frac{P(x)}{Q(x)}$$if the degree of $P$ is (a) less than the degree of $Q$ and(b) greater than the degree of $Q .$
Make a rough sketch of the curve $y=x^{n}(n$ an integer $)$ for the following five cases:(i) $n=0 \quad$ (ii) $n>0, n$ odd(iii) $n>0, n$ even $\quad$ (iv) $n<0, n$ odd(v) $n<0, n$ evenThen use these sketches to find the following limits.$$\text { (a) } \lim _{x \rightarrow 0^{+}} x^{n} \quad \text { (b) } \lim _{x \rightarrow 0^{-}} x^{n}$$$$\text { (c) } \lim _{x \rightarrow \infty} x^{n} \quad \text { (d) } \lim _{x \rightarrow-\infty} x^{n}$$
Find $\lim _{x \rightarrow \infty} f(x)$ if, for all $x>5$$$\frac{4 x-1}{x} < f(x) <\frac{4 x^{2}+3 x}{x^{2}}$$
In the theory of relativity, the mass of a particle with velocity $v$ is$$m=\frac{m_{0}}{\sqrt{1-v^{2} / c^{2}}}$$where $m_{0}$ is the mass of the particle at rest and $c$ is the speed of light. What happens as $v \rightarrow c^{-}$ ?
(a) A tank contains 5000 L of pure water. Brine that contains 30 g of salt per liter of water is pumped into the tank at a rate of 25 $\mathrm{L} / \mathrm{min}$ . Show that the concentration of salt $t$ minutes later (in grams per liter) is$$C(t)=\frac{30 t}{200+t}$$(b) What happens to the concentration as $t \rightarrow \infty ?$
(a) Show that $$\lim _{x \rightarrow \infty} \frac{4 x^{2}-5 x}{2 x^{2}+1}=2$$(b) By graphing the function in part (a) and the line $y=1.9$ on a common screen, find a number $N$ such that$$\text {if} \quad x>N \quad \text { then } \quad \frac{4 x^{2}-5 x}{2 x^{2}+1}>1.9$$What if 1.9 is replaced by 1.99$?$
How close to $-3$ do we have to take $x$ so that$$\frac{1}{(x+3)^{4}}>10,000$$
Prove, using Definition $6,$ that $$\lim _{x \rightarrow-3} \frac{1}{(x+3)^{4}}=\infty$$
Prove that $$\lim _{x \rightarrow-1^{-}} \frac{5}{(x+1)^{3}}=-\infty.$$
For the limit$$\lim _{x \rightarrow \infty} \frac{\sqrt{4 x^{2}+1}}{x+1}=2$$illustrate Definition 7 by finding values of $N$ that correspond to $\varepsilon=0.5$ and $\varepsilon=0.1$
Use a graph to find a number $N$ such that$$\quad \text {if} \quad x>N \quad \text { then } \quad\left|\frac{3 x^{2}+1}{2 x^{2}+x+1}-1.5\right|<0.05$$
For the limit$$\lim _{x \rightarrow \infty} \frac{2 x+1}{\sqrt{x+1}}=\infty$$illustrate Definition 8 by finding a value of $N$ that corresponds to $M=100.$
(a) How large do we have to take $x$ so that $1 / x^{2}<0.0001 ?$(b) Taking $n=2$ in $[5],$ we have the statement$$\lim _{x \rightarrow \infty} \frac{1}{x^{2}}=0$$Prove this directly using Definition 7.
Prove, using Definition $8,$ that $$\lim _{x \rightarrow \infty} x^{3}=\infty.$$
Prove that$$\lim _{x \rightarrow \infty} f(x)=\lim _{t \rightarrow 0^{+}} f(1 / t)$$and$$\lim _{x \rightarrow-\infty} f(x)=\lim _{t \rightarrow 0^{-}} f(1 / t)$$if these limits exist.