To determine

To evaluate:

The value of limit

limx→0⁡1+tan⁡x-1+sin⁡xx3

Answer

The value of limit is limx→0⁡1+tan⁡x-1+sin⁡xx3=14

Explanation

1. Concept:

Use limit of a function

2. Formula:

3.

(i) Standard Formula:

a-ba+b=a2-b2

(ii) Trigonometric formula:

tan⁡x=sin⁡xcos⁡x

(iii) Half Angle Formula:

cos⁡x=1-2sin2 x

⇒1-cos⁡x=2 sin2x2

(iv) Limit of Trigonometric Functions:

limx→0⁡sin⁡xx=1

(v)  Values of Standard Angles:

tan⁡0=0, sin⁡0=0  and cos⁡0=1

3. Given:

limx→0⁡1+tan⁡x-1+sin⁡xx3

4. Calculation:

lim⁡ ⁡ x→0⁡1+tan⁡x-1+sin⁡xx3

Using Rationalization,

=lim⁡ ⁡ x→0⁡1+tan⁡x-1+sin⁡xx31+tan⁡x+1+sin⁡x1+tan⁡x+1+sin⁡x

By Standard Formula

=lim⁡ ⁡ x→0⁡1+tan⁡x-1+sin⁡xx3 (1+tan⁡x+1+sin⁡x)

=lim⁡ ⁡ x→0⁡1+tan⁡x-1-sin⁡xx3 (1+tan⁡x+1+sin⁡x)

=lim⁡ ⁡ x→0⁡tan⁡x-sin⁡xx3 (1+tan⁡x+1+sin⁡x)

By using Standard Formula

=lim⁡ ⁡ x→0⁡sin⁡xcos⁡x-sin⁡xx3 (1+tan⁡x+1+sin⁡x)

=lim⁡ ⁡ x→0⁡sin⁡x-sin⁡xcos⁡xcos⁡xx3 (1+tan⁡x+1+sin⁡x)

=lim⁡ ⁡ x→0⁡sin⁡x-sin⁡xcos⁡xx3 (1+tan⁡x+1+sin⁡x )cos⁡x

=lim⁡ ⁡ x→0⁡sin⁡x(1-cos⁡x)x3 (1+tan⁡x+1+sin⁡x )cos⁡x

=lim⁡ ⁡ x→0⁡sin⁡x2 sin2x2x3 (1+tan⁡x+1+sin⁡x )cos⁡x

=lim⁡ ⁡ x→0⁡2sin⁡xsin2x2x3 (1+tan⁡x+1+sin⁡x )cos⁡x

Divide and Multiply by 2,

=lim⁡ ⁡ x→022⁡2sin⁡xsin2x2x3 (1+tan⁡x+1+sin⁡x )cos⁡x

=lim⁡ ⁡ x→012⁡4sin⁡xsin2x2x3 (1+tan⁡x+1+sin⁡x )cos⁡x

=limx→0⁡sin⁡xxsin2⁡x/2x2/4121(1+tan⁡x+1+sin⁡x )cos⁡x

=limx→0⁡sin⁡xxsin2⁡x/2x2/4121(1+tan⁡x+1+sin⁡x )cos⁡x

By Limit of Trigonometric Functions

=1.1. limx→0⁡121(1+tan⁡x+1+sin⁡x )cos⁡x

=limx→0⁡121(1+tan⁡0+1+sin⁡0 )cos⁡0

By Values of Standard Angles

=limx→0⁡121(1+1 )(1)

=1212

=14

Conclusion:

Therefore, the value of limit is

limx→0⁡1+tan⁡x-1+sin⁡xx3=14