To determine
To evaluate:
The value of limit
limx→01+tanx-1+sinxx3
Answer
The value of limit is limx→01+tanx-1+sinxx3=14
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:
limx→0sinxx=1
(v) Values of Standard Angles:
tan0=0, sin0=0 and cos0=1
3. Given:
limx→01+tanx-1+sinxx3
4. Calculation:
lim x→01+tanx-1+sinxx3
Using Rationalization,
=lim x→01+tanx-1+sinxx31+tanx+1+sinx1+tanx+1+sinx
By Standard Formula
=lim x→01+tanx-1+sinxx3 (1+tanx+1+sinx)
=lim x→01+tanx-1-sinxx3 (1+tanx+1+sinx)
=lim x→0tanx-sinxx3 (1+tanx+1+sinx)
By using Standard Formula
=lim x→0sinxcosx-sinxx3 (1+tanx+1+sinx)
=lim x→0sinx-sinxcosxcosxx3 (1+tanx+1+sinx)
=lim x→0sinx-sinxcosxx3 (1+tanx+1+sinx )cosx
=lim x→0sinx(1-cosx)x3 (1+tanx+1+sinx )cosx
=lim x→0sinx2 sin2x2x3 (1+tanx+1+sinx )cosx
=lim x→02sinxsin2x2x3 (1+tanx+1+sinx )cosx
Divide and Multiply by 2,
=lim x→0222sinxsin2x2x3 (1+tanx+1+sinx )cosx
=lim x→0124sinxsin2x2x3 (1+tanx+1+sinx )cosx
=limx→0sinxxsin2x/2x2/4121(1+tanx+1+sinx )cosx
=limx→0sinxxsin2x/2x2/4121(1+tanx+1+sinx )cosx
By Limit of Trigonometric Functions
=1.1. limx→0121(1+tanx+1+sinx )cosx
=limx→0121(1+tan0+1+sin0 )cos0
By Values of Standard Angles
=limx→0121(1+1 )(1)
=1212
=14
Conclusion:
Therefore, the value of limit is
limx→01+tanx-1+sinxx3=14