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Chapter 1: Q. 31 (page 153)

Find the limit by hand.
$\lim_{x \to \infty} \frac{1}{x} \sin x$

Short Answer

Expert verified

On solving the limit, we get, $0$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$\lim_{x \to \infty} \frac{1}{x} \sin x$

Step 2. Solve the given limit.

Assume $x = \frac{1}{h}$ .

When $x \to \infty$ , then $h \to 0$

Using the limits,

$= \lim_{h \to \infty} h \sin \frac{1}{h} = 0 \sin \frac{1}{0} = 0$

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Most popular questions from this chapter

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 $1$ .

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

Calculate each of the limits:

$\lim_{x \to 3} \frac{\sqrt{x}}{\tan^{- 1} \sqrt{x}}$

For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if localid="1648023101818" $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$

localid="1648023199049" role="math" $\underset{x \to - 1}{l i m} \frac{x^{2} - 2 x - 3}{x + 1} = - 4, ε = 1$

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

$lim_{x \to - 2^{+}} (1 + \sqrt{x + 2}) = 1$ find δ in terms of $ε$ .

Calculate each of the limits:

$\lim_{h \to 0} \frac{{(3 + h)}^{2} - 3^{2}}{h}$ .

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Find the limit by hand.limx→∞1xsinx

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solve the given limit.

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Most popular questions from this chapter

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Find the limit by hand.
$\lim_{x \to \infty} \frac{1}{x} \sin x$