Extend the definition of the following by continuity `f(x)=(1-cos7(x-pi))/(5(x-pi)^2)"at the point"x=pi`

Explanation

Here, we will prove that the righth hand limit and left hand limit is equal for `f(x)` at `x = pi`.

Now, `R.H.L. = Lim_(x->pi^+) (1-cos7(x-pi))/(5(x-pi)^2)`

`= Lim_(h->0) (1-cos7(pi+h-pi))/(5(pi+h-pi)^2)`

`= Lim_(h->0) (1-cos7h)/(5h^2)`

As this is a `0/0`form. So, we will apply `L'`hospital rule to evaluate this limit.

`= Lim_(h->0) (7sin7h)/(10h)`

`= Lim_(h->0) 7/10*7(sin7h)/(7h)`

`=49/10`

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