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Chapter 1: Q. 49 (page 88)

Use tables of values to make educated guesses for each of the limits in Exercises 39–52.
$\lim_{x \to \infty} \frac{x + 1}{x^{2} - 1}$

Short Answer

Expert verified

The value is $0 .$

Step by step solution

Step 1. Given information.

The given function is $\lim_{x \to \infty} \frac{x + 1}{x^{2} - 1}$ .

Step 2. Value of the limit.

Consider the table,

The value approaches to zero as $x \to \infty$

Therefore,

$\lim_{x \to \infty} \frac{x + 1}{x^{2} - 1} = 0$

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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 $x = c$ .

Each function in Exercises 9–12 is discontinuous at some value x = c. Describe the type of discontinuity and any one-sided continuity at x = c, and sketch a possible graph of f.

$\lim_{x \to - 1^{-}} f (x) = 2, \lim_{x \to - 1^{+}} f (x) = 2, f (- 1) = 1 .$

Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) A limit exists if there is some real number that it is equal to.

(b) The limit of $f (x)$ as $x \to c$ is the value $f (c)$ .

(d) The two-sided limit of $f (x)$ as $x \to c$ exists if and only if the left and right limits of $f (x)$ exists as $x \to c$ .

(e) If the graph of $f$ has a vertical asymptote at $x = 5$ , then $\lim_{x \to 5} f (x) = \infty$ .

(f) If $\lim_{x \to 5} f (x) = \infty$ , then the graph of $f$ has a vertical asymptote at $x = 5$ .

(g) If $\lim_{x \to 2} f (x) = \infty$ , then the graph of $f$ has a horizontal asymptote at $x = 2$ .

(h) If $\lim_{x \to \infty} f (x) = 2$ , then the graph of $f$ has a horizontal asymptote at $y = 2$ .

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 $1$ .

$f (x) = x^{- 1 / 2}$

Use the delta-epsilon definition of continuity to argue that f is or is not continuous at the indicated point $x = c$ .

$f (x) = \{\begin{aligned} 2 - x, if x rational \\ x^{2}, if x irrational, \end{aligned} c = 2$

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Use tables of values to make educated guesses for each of the limits in Exercises 39–52.limx→∞x+1x2-1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Value of the limit.

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Most popular questions from this chapter

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Use tables of values to make educated guesses for each of the limits in Exercises 39–52.
$\lim_{x \to \infty} \frac{x + 1}{x^{2} - 1}$