Question

Somebody Please solve this question without using L Hospital rule.

Evaluate if:​

Answer 1

1/2

Explanation:

Given that,

`\lim_(n \to 1) ((1+\cos\pi x)/(\tan^2\pi x))`

Using L Hospital's rule to find it :

`\lim_(n \to a) (f(x))/(g(x))= \lim_(n \to a) (f'(x))/(g'(x))`

We have,

a = 1,
`f(x)=1+\cos\pi x, \ \ g(x)=\tan^2\pi x`

`f'(x)=(d)/(dx)(1+\cos\pi x)\\\\=-\pi \sin \pi x\ .....(1)`

`g'(x)=(d)/(dx)(\tan^2\pi x)\\\\=2\tan\pi x* \sec^2\pi x* \pi\ .....(2)`

From equation (1) and (2) :

`\lim_(n \to 1) (f(x))/(g(x))= \lim_(n \to 1) (-\pi \sin\pi x)/(2\tan \pi x* \sec^2\pi x * \pi)\\\\\lim_(n \to 1) (-\pi \sin\pi x)/(2\tan \pi x* \sec^2\pi x * \pi)\\\\=\lim_(n \to 1) (-1)/(2)*\cos^3\pi x\\\\=(-1)/(2)* \cos^3\pi \\\\=(1)/(2)`

So, the value of the given function is 1/2.