Dear friends, Please read our latest blog post for an important announcement about the website. ❤, The Socratic Team

Lim x->0 (tan x- sin x)/x^3=?

a. 0
b. 1
c. 1/2
d. -1
e. -1/2

a. 0
b. 1
c. 1/2
d. -1
e. -1/2

1 Answer
Write your answer here...
Start with a one sentence answer
Then teach the underlying concepts
Don't copy without citing sources
preview
?

Answer

Write a one sentence answer...

Answer:

Explanation

Explain in detail...

Explanation:

I want someone to double check my answer

Describe your changes (optional) 200

56

This answer has been featured!

Featured answers represent the very best answers the Socratic community can create.

Learn more about featured answers

Nov 9, 2017

Answer:

#lim_(x->0) (tanx-sinx)/x^3 = 1/2#

Explanation:

Transform the function in this way:

#(tanx-sinx)/x^3 = 1/x^3(sinx/cosx-sinx)#

#(tanx-sinx)/x^3 = 1/x^3((sinx-sinxcosx)/cosx)#

#(tanx-sinx)/x^3 = sinx/x^3(1-cosx)/cosx#

#(tanx-sinx)/x^3 = (sinx/x)((1-cosx)/x^2)( 1/cosx)#

We can use now the well known trigonometric limit:

#lim_(x->0) sinx / x = 1#

and using the trigonometric identity:

#sin^2alpha = (1-cos2alpha)/2#

we have:

#lim_(x->0) (1-cosx)/x^2 = lim_(x->0) (2sin^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/cosx = 1/1 = 1#

and we can conclude that:

#lim_(x->0) (tanx-sinx)/x^3 = lim_(x->0) (sinx/x)((1-cosx)/x^2)( 1/cosx) = 1 xx 1/2 xx 1 = 1/2#

graph{(tanx-sinx)/x^3 [-1.25, 1.25, -0.025, 1]}

Was this helpful? Let the contributor know!
1500
Impact of this question
8614 views around the world
You can reuse this answer
Creative Commons License