   Chapter 2.R, Problem 89E

Chapter
Section
Textbook Problem

# Evaluate lim x → 0 1 + tan x − 1 + sin x x 3

To determine

To evaluate:

The value of limit

limx01+tanx-1+sinxx3

Explanation

1. Concept:

Use limit of a function

2. Formula:

3.

(i) Standard Formula:

a-ba+b=a2-b2

(ii) Trigonometric formula:

tanx=sinxcosx

(iii) Half Angle Formula:

cosx=1-2sin2 x

1-cosx=2 sin2x2

(iv) Limit of Trigonometric Functions:

limx0sinxx=1

(v)  Values of Standard Angles:

tan0=0, sin0=0  and cos0=1

3. Given:

limx01+tanx-1+sinxx3

4. Calculation:

lim  x01+tanx-1+sinxx3

Using Rationalization,

=lim  x01+tanx-1+sinxx31+tanx+1+sinx1+tanx+1+sinx

By Standard Formula

=lim  x01+tanx-1+sinxx3 (1+tanx+1+sinx)

=lim  x01+tanx-1-sinxx3 (1+tanx+1+sinx)

=lim  x0tanx-sinxx3 (1+tanx+1+sinx)

By using Standard Formula

=lim  x0sinxcosx-sinxx3 (1+tanx+1+sinx)

=lim  x0sinx-sinxcosxcosxx3 (1+tanx+1+sinx)

=lim  x0sinx-sinxcosxx3 (1+tanx+1+sinx )cosx

=lim  x0sinx(1

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