Chapter 14: Q. 53 (page 883)

In Problems 53-56, use the properties of limits and the facts that
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \lim_{x \to 0} \sin x = 0 \lim_{x \to 0} \cos x = 1$
$\lim_{x \to 0} \frac{\tan x}{x}$

Short Answer

Expert verified

The limit equals $1$

Step by step solution

Given information

Given function $\lim_{x \to 0} \frac{\tan x}{x}$

Evaluating the limit by breaking into sine and cosine components

Calculating, we get

$\lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} (\frac{\sin x}{x} \cdot \frac{1}{\cos x}) \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} (\frac{\sin x}{x}) \cdot \lim_{x \to 0} (\frac{1}{\cos x}) \lim_{x \to 0} \frac{\tan x}{x} = 1 \cdot \frac{1}{1} \lim_{x \to 0} \frac{\tan x}{x} = 1$

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In Problems 53-56, use the properties of limits and the facts that
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \lim_{x \to 0} \sin x = 0 \lim_{x \to 0} \cos x = 1$
$\lim_{x \to 0} \frac{\tan x}{x}$

Short Answer

Step by step solution

Given information

Evaluating the limit by breaking into sine and cosine components

One App. One Place for Learning.

Most popular questions from this chapter

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In Problems 53-56, use the properties of limits and the facts thatlimx→0sinxx=1limx→0cosx-1x=0limx→0sinx=0limx→0cosx=1limx→0tanxx

Short Answer

Step by step solution

Given information

Evaluating the limit by breaking into sine and cosine components

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Mechanics Maths

Discrete Mathematics

Geometry

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 53-56, use the properties of limits and the facts that
$\lim_{x \to 0} \frac{\sin x}{x} = 1 \lim_{x \to 0} \frac{\cos x - 1}{x} = 0 \lim_{x \to 0} \sin x = 0 \lim_{x \to 0} \cos x = 1$
$\lim_{x \to 0} \frac{\tan x}{x}$