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Chapter 1: Q. 10 (page 153)

Consider the limit expression
$\lim_{x \to \infty} \ln (\frac{x - 1}{1 - 3 x^{2}})$
Calculate the limit.

Short Answer

Expert verified

The limit of the expression is $\infty$ .

Step by step solution

01

Step 1. Given

The given expression is

$\lim_{x \to \infty} \ln (\frac{x - 1}{1 - 3 x^{2}})$

02

Step 2. Calculation

The evaluated expression is as follow:

$|\lim_{x \to \infty} \ln (\frac{x - 1}{1 - 3 x^{2}})| = \lim_{x \to \infty} \ln (\frac{x^{2} (\frac{1}{x} - \frac{1}{x^{2}})}{x^{2} (\frac{1}{x^{2}} - 3)}) = \lim_{x \to \infty} \ln (\frac{\frac{1}{x} - \frac{1}{x^{2}}}{\frac{1}{x^{2}} - 3}) = \ln (\frac{\frac{1}{\infty} - \frac{1}{\infty^{2}}}{\frac{1}{\infty^{2}} - 3}) = \ln (\frac{0 - 0}{0 - 3}) = \ln (0) = \infty$

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Most popular questions from this chapter

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.

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

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.

$\lim_{x \to \infty} (x^{2} + 2) = \infty$ ,find $N$ in terms of $M$ .

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x \to 8} (3 x - 11) = 13$ .

Write delta-epsilon proofs for each of the limit statements $\lim_{x \to c} f (x) = L$ in Exercises $47 - 60$ .

$\lim_{x \to 1} (2 x + 4) = 6$ .

In Exercises 39–44, use Theorem 1.16 and left and right limits to determine whether each function f is continuous at its break point(s). For each discontinuity of f, describe the type of discontinuity and any one-sided discontinuity.

$f (x) = \{\begin{matrix} x^{2}, i f x < 2 \\ 4, i f x = 2 \\ 2 x + 1, i f x > 2 \end{matrix} .$

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