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Chapter 1: Q. 52 (page 88)

Use tables of values to make educated guesses for each of the limits in Exercises 39–52.
$\lim_{x \to \infty} \sin (\frac{1}{x})$

Short Answer

Expert verified

The value of $\lim_{x \to \infty} \sin (\frac{1}{x}) = 0 .$

Step by step solution

01

Step 1. Given Information.

The given function is $\lim_{x \to \infty} \sin (\frac{1}{x})$ .

02

Step 2. Value of the limit.

Consider the table,

$x$	$10$	$100$	$1000$
$\sin (\frac{1}{x})$	$0.0841$	$0.0084$	$0.00084$

The value of $\sin (\frac{1}{x})$ approaches to zero as $x \to \infty .$

Therefore,

$\lim_{x \to \infty} \sin (\frac{1}{x}) = 0$

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Most popular questions from this chapter

Use what you know about one-sided limits to prove that a function $f$ is continuous at a point $x = c$ if and only if it is both left and right continuous at $x = c$ .

Find a formula for the cost $C (r)$ of producing a gourmet soup can with radius $r$ and height $5$ inches, and answer the following questions:

What is the radius of a can that is $5$ inches tall and costs $30$ cents to produce?
Your manager wants you to produce $5$ -inch-tall cans that cost between $20$ and $40$ cents. Write this requirement as an absolute value inequality.
What range of radii would satisfy your manager? Write an absolute value inequality whose solution set lies inside this range of radii.

For each limit in Exercises 33–38, either use continuity to calculate the limit or explain why Theorem 1.16 does not apply.

$\lim_{x \to - 1} x .$

Write a delta–epsilon proof that proves that $f$ is continuous on its domain. In each case, you will need to assume that δ is less than or equal to $1$ .

$f (x) = x^{2}$

For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.

$\lim_{x \to 1^{-}} \frac{1}{1 - x} = \infty$ , find $δ$ in terms of $M$ .

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