Evaluate `lim_(xto7//2) (2x^(2)-9x+8)^(cot(2x-7)).`

Options

A. both `underset(xtoa)limf(x)` and `underset(xtoa)limg(x)` must exist
B. `underset(xtoa)limf(x)` need not exist but `underset(xtoa)limg(x)` exists
C. neither `underset(xtoa)limf(x)` nor `underset(xtoa)limg(x)` may exist
D. `underset(xtoa)limf(x)` exists but `underset(xtoa)limg(x)` need not exist

Answer

`e^(5//2)`

Explanation

Given limit takes `1^(oo)` form: Therefore,
`L=underset(xto7//2)lim(2x^(2)-9x+8)^(cot(2x-7))`
`=underset(xto7//2)lim((2x-7)(x-1)+1)^(cot(2x-7))`
`=e^(underset(xtox_(7//2))lim((2x-7)(x-1))cot(2x-7))`
`=e^(underset(xtox_(7//2))lim((2x-7)(x-1))/(tan(2x-7)))`
`=e^(5//2)`