Limits

Calculate each limit in Exercises 35–80. 

$\underset{x\to \infty }{lim}\frac{x^{{\displaystyle \raisebox{1ex}{$7$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}-x^{{\displaystyle \raisebox{1ex}{$8$}\!\left/ \!\raisebox{-1ex}{$3$}\right.}}}{x^2}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to \infty }{lim}\frac{4^x-3^x}{5^x}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to \infty }{lim}\frac{2^x-4^{-x}}{3^x}$

Calculate each limit in Exercises 35–80.  

$\underset{x\to -\infty }{lim}\frac{3^x-5^x}{4^x}$

Calculate each limit in Exercises 35–80.  

$\underset{x\to \infty }{lim}\frac{4\left(3^x\right)}{2+3^x}$

Calculate each limit in Exercises 35–80.  

$\underset{x\to \infty }{lim}\frac{2e^{1.5x}}{3e^{2x}+e^{1.5x}}$

Calculate each limit in Exercises 35–80.   

$\underset{x\to \infty }{lim}\frac{1-5e^{2x}}{3e^x+4e^{2x}}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 3^{+}}{lim}\ln (x^2-9)$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0^{+}}{lim}\ln \left(\frac{1}{x}\right)$

Calculate each limit in Exercises 35–80. 

$\underset{x\to \infty }{lim}\left(\ln x^2-\ln (2x+1)\right)$

Calculate each limit in Exercises 35–80. 

$\underset{x\to \infty }{lim}(\ln 3x-\ln 2x)$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{1-\cos 2x}{7x}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{\sin 3x}{5x}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{x}{1-\cos x}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{3\sin x+x}{x}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{{\sin }^{2}3x}{x^3-x}$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\left(\frac{\sin \left(3x^2\right)}{x^3-x}\right)$

Calculate the limit.

$\underset{x\to 0^{+}}{\lim } \frac{x^{2}csc 3x}{1-\cos 2x}$

Find the limit.

$\underset{x\to 0}{\lim }\frac{x^{2}cot x}{\sin x}$

Find the limit.

$\underset{x\to 0}{\lim }\frac{sec x \tan x}{x}$

Find the limit.

$\underset{x\to 0}{\lim }3x^{2 }co{t}^{2}x$

Find the limit.

$\underset{x\to 0}{\lim }{\left(1+x\right)}^{\frac{2}{x}}$

Find the limit.

$\underset{x\to 0}{\lim }{\left(1+2x\right)}^{\frac{3}{x}}$

Find the limit.

$\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^{3x}$

Find the limit.

$\underset{x\to \infty }{\lim }{\left(1-\frac{5}{x}\right)}^x$

Calculate each limit in Exercises 35–80. 

$\underset{x\to 1}{\lim } \frac{x-1}{x^2-2x+1}$

In 1960, H. von Foerster suggested that the human population could be measured by the function $P\left(t\right)=\frac{179\times {10}^9}{{\left(2027-t\right)}^{0.99}}$

The time t is measured in years, where t = 1 corresponds to the year 1 a.d., 

t = 1973 corresponds to the year 1973 a.d., and so on. (We saw this “doomsday model” for population in Problem 77 of Section 1.1, on page 89.) Use limit

techniques to calculate $\underset{t\to {2027}^{-}}{\lim }P\left(t\right)$. What does this limit mean in real-world terms?

Suppose instead we consider the population model

$Q\left(t\right)=\frac{44\times {10}^{10}}{1+{\left(2027-t\right)}^{{\displaystyle \frac{4}{3}}}}$

with t measured in years as in the previous problem.

(a) Use limit techniques to calculate $\underset{t\to \infty }{\lim }Q\left(t\right)$. What does this limit mean in real–world terms? What happens in this model in the year 2027?

(b) Use calculator graphs to compare the population models in this exercise with those in the previous exercise. Describe the long–term population growth

scenarios that are suggested by these models.

Consider a mass hanging from the ceiling at the end of a spring. If you pull down on the mass and let go, it will oscillate up and down according to the equation

$s\left(t\right)=A\sin \left(\sqrt{\frac{k}{m}}t\right)+B\cos \left(\sqrt{\frac{k}{m}}t\right)$

where $s\left(t\right)$ is the distance of the mass from its equilibrium position, m is the mass of the bob on the end of the spring, and k is a "spring coefficient" that measures how tight or stiff the spring is. The constants A and B depend on initial conditions - specifically, how far you pull down the mass $\left(s_0\right)$ and the velocity at which you release the mass $\left(v_0\right)$. This equation does not take into effect any friction due to air resistance.

1. Determine whether or not the limit of $s\left(t\right)$ as $t\to \infty$ exists. What does this say about the long-term behavior of the mass on the end of the spring?
2. Explain how this limit relates to the fact that the equation for $s\left(t\right)$ does not take friction due to air resistance into account.
3. Suppose the bob at the end of the spring has a mass of $2$ grams and that the coefficient for the spring is $k=9$. Suppose also that the spring is released in such a way that $A=\sqrt{2}$ and $B=2$. Use a graphing utility to graph the function $s\left(t\right)$ that describes the distance of the mass from its equilibrium position. Use your graph to support your answer to part (a).

In the previous exercise we gave an equation describing spring motion without air resistance. If we take into account friction due to air resistance, the mass will oscillate up and down according to the equation

$s\left(t\right)=e^{(-f/2m)t}\left(A\sin \left(\frac{\sqrt{4km-f^2}}{2m}t\right)\right.\left.+B\cos \left(\frac{\sqrt{4km-f^2}}{2m}t\right)\right)$

where m, k, A, and B are the constants described in Problem $83$ and f is a positive "friction coefficient" that measures the amount of friction due to air resistance.

1. Find the limit of $s\left(t\right)$ as $t\to \infty$. What does this say about the long-term behavior of the mass on the end of the spring?
2. Explain how this limit relates to the fact that the new equation for $s\left(t\right)$ does take friction due to air resistance into account.
3. Suppose the bob at the end of the spring has a mass of $2$ grams, the coefficient for the spring is $k=9$, and the friction coefficient is $f=6$. Suppose also that the spring is released in such a way that $A=4$ and $B=2$. Use a graphing utility to graph the function $s\left(t\right)$ that describes the distance of the mass from its equilibrium position. Use your graph to support your answer to part (a).

A limit representing an instantaneous rate of change: After t seconds, a bowling ball dropped from $350$feet has height $h\left(t\right)=350-16t^2,$measured in feet.

Calculate the average rate of change of the height of the bowling ball from $t=3$to $t=3+h$seconds in the cases where h is equal to $0.5, 0.25, 0.1,$and$0.01.$

A limit representing an instantaneous rate of change: After t seconds, a bowling ball dropped from $350$feet has height $h\left(t\right)=350-16t^2,$measured in feet.

Write down a formula for the average rate of change of the height of the bowling ball from time $t=3$to time $t=3+h,$assuming that $h>0.$The only letter in your formula should be h.

A limit representing an instantaneous rate of change: After t seconds, a bowling ball dropped from $350$feet has height $h\left(t\right)=350-16t^2,$measured in feet.

Take the limit as $h\to 0^{+}$of the formula you found for average rate of change in the previous problem. What does this limit represent in real-world terms?

In this section we learned that e can be thought of as the following limit:

$\underset{h\to 0}{\lim }{\left(1+h\right)}^{\frac{1}{h}}=e.$

In the following exercise, you will investigate the convergence of this limit and also get a preview of the Taylor series, which we will see in Chapter 8.

Use the substitution $n=\frac{1}{h}$to show that the preceding limit statement is equivalent to the limit statement

$\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n=e.$

The Binomial Theorem says that an expression of the form ${\left(a+b\right)}^n$can be expanded to $\left(\begin{array}{c}n\\ 0\end{array}\right)a^n{b}^0+\left(\begin{array}{c}n\\ 1\end{array}\right)a^{n-1}{b}^1+\left(\begin{array}{c}n\\ 2\end{array}\right)a^{n-1}{b}^2+\cdots +\left(\begin{array}{c}n\\ n\end{array}\right)x^0{y}^n,$where for any $0\le k\le n,$the symbol $\left({{}_k}^n\right)$is equal to $\frac{n!}{k!\left(n-k\right)!}.$Here $n!$is n factorial, the product of the integers from $1$to n. By convention we set $0!=1.$Apply this expansion to the expression ${\left(1+\frac{1}{n}\right)}^n.$

Show that as $n\to \infty$we would expect the preceding expansion to approach

$1+\frac{1}{1!}+\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\frac{1}{5!}+\cdots .$

Use a calculator to find the sum of the first six terms of the sum from the previous problem, and compare this sum with your calculator’s best approximation of the number e.

Use limits to prove that the limits of a polynomial $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+a_{1}x+a_0$ are the same as the limits of its leading term $a_n{x}^n$ as $x\to \infty$ and as $x\to -\infty$. (Hint: Show that $\underset{x\to \infty }{\lim }f\left(x\right)$ is equal to $\underset{x\to \infty }{\lim }{a}_n{x}^n$ by factoring out $a_n{x}^n$ from $f\left(x\right)$.)

Use limit techniques to prove that a rational function $f\left(x\right)=\frac{p\left(x\right)}{q\left(x\right)}$ will have,

1. a horizontal asymptote at $y = 0$, if the degree of p(x) is less than the degree of q(x);
2. a horizontal asymptote at $y=\frac{a_n}{b_m}$, where $a_n$ and $b_n$are the leading terms of p(x) and q(x), respectively, if p(x) and q(x) have the same degree;
3. no horizontal asymptote, if the degree of p(x) is greater than the degree of q(x).

Prove the second part of Theorem $1.29$: If $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is of the form $\frac{1}{0^{-}}$, then $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=-\infty$.

Prove the second part of Theorem $1.30$: If $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is of the form $\frac{1}{-\infty }$, then $\underset{x\to \infty }{\lim }\frac{f\left(x\right)}{g\left(x\right)}=0$.

Prove that the sum rule for limits also applies for limits as $x\to \infty$ : If $\underset{x\to \infty }{\lim }f\left(x\right)=L$ and $\underset{x\to \infty }{\lim }g\left(x\right)=M$, then $\underset{x\to \infty }{\lim }\left(f\right(x)+g(x\left)\right)=L+M.$

Prove the second part of Theorem$1.31$(a): If $k>0$, then $\underset{x\to \infty }{\lim }{x}^{-k}=0$.

Prove the $k=1$ case of the first part of Theorem 1.31(b): that $\underset{x\to \infty }{\lim }{e}^x=\infty$. (Hint: Given $M>0$, choose $N=\ln M$. Then if $x>N=\ln M$, we must have $x=\ln M+c$ for some positive number c. Use this to show that $e^x>M$.)

Prove the $k=1$ case of the second part of Theorem $1.31$(b): that $\underset{x\to \infty }{\lim }{e}^{-x}=0$.

Prove that $\underset{\theta \to 0}{\lim }\frac{1-\cos \theta }{\theta }=0$ by using the double-angle identity $\cos 2\theta =1-2{\sin }^2\theta$ and the other special trigonometric limit $\underset{\theta \to 0}{\lim }\frac{\sin \theta }{\theta }=1$.

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible. 

the intuitive meaning of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=L,\underset{x\to c^{-}}{\lim }f\left(x\right)=L$and $\underset{x\to c^{+}}{\lim }f\left(x\right)=L$

Fill in the blanks to complete each of the following theorem statements: 

If $\underset{x\to c}{\lim }f\left(x\right)=L$and$\underset{x\to c}{\lim }f\left(x\right)=M$, then                  . 

Give precise mathematical definitions or descriptions of each of the concepts that follow. Then illustrate the definition with a graph or algebraic example, if possible. 

the intuitive meaning of the limit statements $\underset{x\to c}{\lim }f\left(x\right)=\infty$ and $\underset{x\to \infty }{\lim }f\left(x\right)=L$

Fill in the blanks to complete each of the following theorem statements: 

$\underset{x\to c}{\lim }f\left(x\right)=L$if and only if $\underset{x\to c^{-}}{\lim }f\left(x\right)=\underline{ }$and$\underset{x\to c^{+}}{\lim }f\left(x\right)=\underline{ }.$