Lim x tends to 0 then (tanx-sinx)/x×x×x
Answers
Answer:
Answer:
lim
x
→
0
tan
x
−
sin
x
x
3
=
1
2
Explanation:
Transform the function in this way:
tan
x
−
sin
x
x
3
=
1
x
3
(
sin
x
cos
x
−
sin
x
)
tan
x
−
sin
x
x
3
=
1
x
3
(
sin
x
−
sin
x
cos
x
cos
x
)
tan
x
−
sin
x
x
3
=
sin
x
x
3
1
−
cos
x
cos
x
tan
x
−
sin
x
x
3
=
(
sin
x
x
)
(
1
−
cos
x
x
2
)
(
1
cos
x
)
We can use now the well known trigonometric limit:
lim
x
→
0
sin
x
x
=
1
and using the trigonometric identity:
sin
2
α
=
1
−
cos
2
α
2
we have:
lim
x
→
0
1
−
cos
x
x
2
=
lim
x
→
0
2
sin
2
(
x
2
)
x
2
=
1
2
lim
x
→
0
⎛
⎜
⎝
sin
(
x
2
)
x
2
⎞
⎟
⎠
2
=
1
2
While the third function is continuous so:
lim
x
→
0
1
cos
x
=
1
1
=
1
and we can conclude that:
lim
x
→
0
tan
x
−
sin
x
x
3
=
lim
x
→
0
(
sin
x
x
)
(
1
−
cos
x
x
2
)
(
1
cos
x
)
=
1
×
1
2
×
1
=
1
2
graph{(tanx-sinx)/x^3 [-1.25, 1.25, -0.025, 1]}