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Chapter 1: Q. 53 (page 88)

Sketch graphs by hand and use them to make approximations for each of the limits in Exercises 53–66. If a two-sided limit does not exist, describe the one-sided limits.
$\lim_{x \to 0} \frac{1}{x}$

Short Answer

Expert verified

The limit value does not exist.

Step by step solution

Step 1. Given information.

The given function is $\lim_{x \to 0} \frac{1}{x}$ .

Step 2. Graph of the function.

The graph of the function is

We can see,

$A s x \to 0, f (x) \to - \infty (L e f t) A s x \to 0, f (x) \to \infty (R i g h t)$

Therefore, the limit does not exist.

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Most popular questions from this chapter

Sketch the graph of a function f described in Exercises 27–32, if possible. If it is not possible, explain why not.

f is left continuous at x = 1 and right continuous at x = 1, but is not continuous at x = 1, and f(1) = −2.

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 $x = c$ .

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.

$\lim_{x \to \infty} \frac{x - 1}{x} = 1$ , find $N$ in terms of $ε$ .

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 $1$ .

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

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 $1$ .

$f (x) = x^{3}$

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Sketch graphs by hand and use them to make approximations for each of the limits in Exercises 53–66. If a two-sided limit does not exist, describe the one-sided limits.limx→01x

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Graph of the function.

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Most popular questions from this chapter

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Sketch graphs by hand and use them to make approximations for each of the limits in Exercises 53–66. If a two-sided limit does not exist, describe the one-sided limits.
$\lim_{x \to 0} \frac{1}{x}$