Evaluation of limit.

**rishilaish** · October 9th 2011, 04:28 AM

The problem is to evaluate the limit using la'hospital rule..
$\lim_{x\rightarrow a} \left ( 2-\frac{x}{a} \right )^{tan\frac{\pi x}{2a}$

Soln.

Let, y = $\lim_{x\rightarrow a} \left ( 2-\frac{x}{a} \right )^{tan\frac{\pi x}{2a}}$
logy = $\lim_{x\rightarrow a} tan\frac{\pi x}{2a}\log \left (2-\frac{x}{a} \right )$
after applying La'Hospital rule, i gt this,
logy = $\frac{\pi}{2} \lim_{x\rightarrow a} sec^{2}\frac{\pi x}{2a}.tan\frac{\pi x}{2a}$

....i gt stuck in this step coz i gt repeatedly 0/0 form..n pliz suggest how do i write all these mathematical equations in faster and efficeint way, is there any software to make it possible???

**Siron** · October 9th 2011, 05:54 AM

You can rewrite:
$\lim_{x \to a} \left(2-\frac{x}{a}\right)^{\tan\left(\frac{\pi x}{2a}\right)}=e^{\lim_{x \to a}\frac{\ln\left(2-\frac{x}{a}\right)}{\tan\left(\frac{\pi x}{2a}\right)}$

And you can write $\tan\left(\frac{\pi x}{2a}\right)=\frac{\sin\left(\frac{\pi x}{2a}\right)}{\cos\left(\frac{\pi x}{2a}\right)}$

Therefore you get:
$e^{\lim_{x \to a}\frac{\cos\left(\frac{\pi x}{2a}\right)\ln\left(2-\frac{x}{a}\right)}{\sin\left(\frac{\pi x}{2a}\right)}$

Can you compute the limit?

**rishilaish** · October 9th 2011, 06:56 AM

m confuse how $tan\frac{\pi x}{2a}$ comes in denominator...

**SammyS** · October 9th 2011, 07:26 AM

Originally Posted by rishilaish

The problem is to evaluate the limit using la'hospital rule..
$\lim_{x\rightarrow a} \left ( 2-\frac{x}{a} \right )^{tan\frac{\pi x}{2a}$

Soln.

log(y) = $\lim_{x\rightarrow a} tan\frac{\pi x}{2a}\log \left (2-\frac{x}{a} \right )$
after applying L'Hospital rule, i gt this,

You can't apply L'Hôpital's Rule directly to a limit of the form (±∞)(0) .

**rishilaish** · October 9th 2011, 07:49 AM

Originally Posted by SammyS

You can't apply L'Hôpital's Rule directly to a limit of the form (±∞)(0) .

No...i hv reduced it into 0/0 form and after applying l'hospital rule...still getting 0/0 form...

**Siron** · October 9th 2011, 09:49 AM

Sorry, my bad, it has to be:
$e^{\lim_{x \to a}\frac{\sin\left(\frac{\pi x}{2a}\right)\ln\left(2-\frac{x}{a}\right)}{\cos\left(\frac{\pi x}{2a}\right)}$

Now you can apply l'Hopital's rule.

**rishilaish** · October 9th 2011, 10:48 AM

siron..pliz help me out wiht...Evaluation of limit(2)

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