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Chapter 1: Q. 39 (page 149)

Calculate each limit in Exercises 35-80.
$\lim_{x \to \infty} (\sqrt{x} - x)$

Short Answer

Expert verified

The limit of the function is $\lim_{x \to \infty} (\sqrt{x} - x) = - \infty$ .

Step by step solution

01

Step 1. Given information.

We have:

$\lim_{x \to \infty} (\sqrt{x} - x)$

02

Step 2. Find the limits.

The function is $\lim_{x \to \infty} (\sqrt{x} - x) .$

Apply the following algebraic property: $a + b = a (1 + \frac{b}{a})$

$= \lim_{x \to \infty} (x (\frac{1}{\sqrt{x}} - 1)) \lim_{x \to a} [f (x) \cdot g (x)] = \lim_{x \to a} f (x) \cdot \lim_{x \to a} g (x) \lim_{x \to a} [f (x) \cdot g (x)] = \lim_{x \to a} f (x) \cdot \lim_{x \to a} g (x) = \infty \cdot (- 1) = - \infty$

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Most popular questions from this chapter

For each functionf graphed in Exercises23–26, describe the intervals on whichf is continuous. For each discontinuity off, describe the type of discontinuity and any one-sided continuity. Justify your answers about discontinuities with limit statements.

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

$\lim_{x \to 0} x^{- 3} .$

For each of the following sign charts, sketch the graph of a function f that has the indicated signs, zeros, and discontinuities:

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

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

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .

$\lim_{x \to 3} \frac{1}{x} = \frac{1}{3}; you may assume δ \leq 1$

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