1.

If lim_(x to 0) (2^(sin x^(2)) - 2^(tan x^2) - 2x^(7))/(x^n) exists finitely and equal to a non-zero number then n is

Answer»

3
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7

Solution :`lim_(x to 0)(2^(sin x^(2)) - 2^(tan x^2) - 2X^(7))/(x^n) = lim_(x to 0) (2^(tan x^2)(2^(sin x^(2)tan x^2) -1)- 2x^(7))/(x^n)`
=`lim_(x to 0) ((2^(tan x^2)(2^(sin x^(2)tan x^2) -1))/(SINX^(2)-tanx^(2)) xx (sin x^(2) - tan x^(2))- 2x^(7))/(x^n)`
=`lim_(x to 0) 1/(x^n)[(2^(tan x^2)(2^(sin x^(2)tan x^2) -1))/(sinx^(2)-tanx^(2)) ((x^(2) - (x^6)/(3 !) + (x^10)/(5!)....)-(x^(2) + (x^6)/3 + (2x^(10))/(15)+....))-2x^(7)]`
`= lim_(x to 0) (x^6)/(x^n)[(2^(tan x^2)(2^(sin x^(2)tan x^2) -1))/((sinx^(2)-tanx^(2))) (-1/2 + "term containing"x)-2x^(1)]`
`implies n = 6`.


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