# Need help evaluating a limit

## Homework Statement

Evaluate the limit ##\lim_{x\to0} \frac{1}{x}(\frac{1}{tanx}-\frac{1}{x}) ## using Taylor's formula. (Hint: ##\frac{1}{1+c}=\frac{1-c^2+c^2}{1+c} ## may be useful)

## The Attempt at a Solution

I began by substituting ##tanx## with ##x+\frac{x^3}{3}+x^3ε(x)##, where ε tends to zero as x approaches 0.

##\frac{1}{x}(\frac{1}{tanx}-\frac{1}{x})=\frac{1}{x}(\frac{x-tanx}{xtanx})=\frac{1}{x}(\frac{x-x-\frac{x^3}{3}-x^3ε(x))}{x(x+\frac{x^3}{3}+x^3ε(x))})=\frac{1}{x}(\frac{-\frac{x^3}{3}-x^3ε(x))}{x(x+\frac{x^3}{3}+x^3ε(x))})=\frac{-\frac{x^3}{3}-x^3ε(x))}{x^2(x+\frac{x^3}{3}+x^3ε(x))}=\frac{-\frac{x^3}{3}-x^3ε(x))}{x^3+\frac{x^4}{3}+x^4ε(x))}=\frac{-\frac{1}{3}-ε(x))}{1+\frac{x}{3}+xε(x))}## →-1/3 as x→0

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stevendaryl
Staff Emeritus

## Homework Statement

Evaluate the limit ##\lim_{x\to0} \frac{1}{x}(\frac{1}{tanx}-\frac{1}{x}) ## using Taylor's formula. (Hint: ##\frac{1}{1+c}=\frac{1-c^2+c^2}{1+c} ## may be useful)

## The Attempt at a Solution

substituting ##tanx## with ##x-\frac{x^3}{3}+x^3ε(x)##, where ε tends to zero as x approaches 0.

##\frac{1}{x}(\frac{1}{tanx}-\frac{1}{x})=\frac{1}{x}(\frac{x-tanx}{xtanx})=\frac{1}{x}(\frac{x-x-\frac{x^3}{3}+x^3ε(x))}{x(x-\frac{x^3}{3}+x^3ε(x))})=\frac{1}{x}(\frac{\frac{x^3}{3}+x^3ε(x))}{x(x-\frac{x^3}{3}+x^3ε(x))})##
You're not far from the answer. After cancellations, you have: $\frac{\frac{x^3}{3} + ...}{x^3 + ...}$, where $...$ represents higher-order terms. If you ignore the higher-order terms, what do you get?

You're not far from the answer. After cancellations, you have: $\frac{\frac{x^3}{3} + ...}{x^3 + ...}$, where $...$ represents higher-order terms. If you ignore the higher-order terms, what do you get?
It's 1/3. But where to use the hint I am given?

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Samy_A
Homework Helper

stevendaryl
Staff Emeritus
It's 1/3. But where to use the hint I am given?
I don't get that, either.

You are right. It's a typo.

PeroK
Homework Helper
Gold Member
2020 Award

## Homework Statement

Evaluate the limit ##\lim_{x\to0} \frac{1}{x}(\frac{1}{tanx}-\frac{1}{x}) ## using Taylor's formula. (Hint: ##\frac{1}{1+c}=\frac{1-c^2+c^2}{1+c} ## may be useful)

## The Attempt at a Solution

I began by substituting ##tanx## with ##x-\frac{x^3}{3}+x^3ε(x)##, where ε tends to zero as x approaches 0.

##\frac{1}{x}(\frac{1}{tanx}-\frac{1}{x})=\frac{1}{x}(\frac{x-tanx}{xtanx})=\frac{1}{x}(\frac{x-x-\frac{x^3}{3}+x^3ε(x))}{x(x-\frac{x^3}{3}+x^3ε(x))})=\frac{1}{x}(\frac{\frac{x^3}{3}+x^3ε(x))}{x(x-\frac{x^3}{3}+x^3ε(x))})=\frac{\frac{x^3}{3}+x^3ε(x))}{x^2(x-\frac{x^3}{3}+x^3ε(x))}=\frac{\frac{x^3}{3}+x^3ε(x))}{x^3-\frac{x^4}{3}+x^4ε(x))}##??
I think you've got the wrong series for ##tan(x)##. Check your coefficients. Also, if you are going to use the Taylor series, you should use the series for ##1/tan(x)## by applying the binomial expansion to the series for ##tan(x)## or using the series for ##cot(x)##.

That said, this one looks tailor-made(!) for L'Hopital, using ##tan = sin/cos##.

stevendaryl
Staff Emeritus
No, I think he started with the wrong expansion for tan(x). It should be $tan(x) = x + \frac{x^3}{3} + ...$ not $x - \frac{x^3}{3}$

Samy_A
Homework Helper
No, I think he started with the wrong expansion for tan(x). It should be $tan(x) = x + \frac{x^3}{3} + ...$ not $x - \frac{x^3}{3}$
Yes, he has a wrong Taylor series. In the denominator, that should have been ##+\frac{x^3}{3}##. But in the numerator, he expands ##x-\tan x##, so the ##\frac{x^3}{3}## gets a minus sign.
When taking the limit, the ##x³## term resulting from the ##\tan## series in the denominator is not important, but the one in the numerator is.
The correct limit is ##-\frac13##.

PeroK
Homework Helper
Gold Member
2020 Award
My advice would be to do it the easy way using L'Hopital so you know what the answer is, then do it the hard way using Taylor series

Samy_A
Homework Helper
My advice would be to do it the easy way using L'Hopital so you know what the answer is, then do it the hard way using Taylor series
Sure, the hints he got are strange. This is, as you said, tailor-made(!) for L'Hopital.

##\frac{1}{x}(\frac{1}{tanx}-\frac{1}{x})=\frac{1}{x}(\frac{x-tanx}{xtanx})=\frac{1}{x}(\frac{x-x-\frac{x^3}{3}-x^3ε(x))}{x(x+\frac{x^3}{3}+x^3ε(x))})=...=\frac{-\frac{x^3}{3}-x^3ε(x))}{x^3+\frac{x^4}{3}+x^4ε(x))}=\frac{-\frac{1}{3}-ε(x))}{1+\frac{x}{3}+xε(x))}## →-1/3 as x→0

Looks like I didn't need the hint. However, I think I'm supposed to derive the Taylor formula I used for the problem. We have covered expansions for sinx and cos x in class. So ##tanx=\frac{sinx}{cosx}=\frac{x-\frac{x^3}{3!}+\frac{x^5}{5!}-...}{1-\frac{x^2}{2!}+\frac{x^4}{4!}-...}=##

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PeroK
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2020 Award
##\frac{1}{x}(\frac{1}{tanx}-\frac{1}{x})=\frac{1}{x}(\frac{x-tanx}{xtanx})=\frac{1}{x}(\frac{x-x-\frac{x^3}{3}-x^3ε(x))}{x(x+\frac{x^3}{3}+x^3ε(x))})=...=\frac{-\frac{x^3}{3}-x^3ε(x))}{x^3+\frac{x^4}{3}+x^4ε(x))}=\frac{-\frac{1}{3}-ε(x))}{1+\frac{x}{3}+xε(x))}## →-1/3 as x→0

Looks like I didn't need the hint. However, I think I'm supposed to derive the Taylor formula I used for the problem. We have covered expansions for sinx and cos x in class. So ##tanx=\frac{sinx}{cosx}=\frac{x-\frac{x^3}{3!}+\frac{x^5}{5!}-...}{1-\frac{x^2}{2!}+\frac{x^4}{4!}-...}=##
I would go for: ##1/tanx = cosx/sinx##

You might as well make things easy for yourself!

Use the binomial for ##1/sinx = (x-\frac{x^3}{3!}+\frac{x^5}{5!}- \dots)^{-1} = (1/x)(1-\frac{x^2}{3!}+\frac{x^4}{5!}- \dots)^{-1}##

I actually derived the expansion for ##\tan x## from the definition of taylor formula. ##f'(x)=D\tan x=1+\tan^2 x##, ##f'(0)=1##
##f''(x)=2\tan x(1+\tan^2 x)##, ##f''(0)=0## etc. and got ##\tan x=x+\frac{x^3}{3}+x^3ε(x)##. I think this is maybe shorter and nicer than deriving it by long division.

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SammyS
Staff Emeritus
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I would go for: ##1/ \tan x = \cos x / \sin x##
I thought I had posted something on this a day or two ago. Using the first of Perok's suggestions, then common denominator, etc. :

##\displaystyle \
\frac{1}{x}\left(\frac{1}{\tan x}-\frac{1}{x}\right)
\ ##

##\displaystyle \
=\frac{x \cos x - \sin x}{x^2\sin x}
\ ##

Now use the Taylor expansions for sin x and cos x .