Yo sup??
For our convenience let h=x+1
therefore
when x tends to -1, h tends to 0
hence we can rewrite it as
[tex]\lim_{h \to \ 0 } (cos(h))^{(cot(h^2 )}[/tex]
This inequality is of the form 1∞
We will now apply the formula
[tex]e^(^g^(^x^)^(^f^(^x^)^-^1^)^)[/tex]
plugging in the values of g(x) and f(x)
[tex]e^{lim_{h \to \ 0}{(cot(h)^2(cos(h)-1))}[/tex]
express coth² as cosh²/sinh² and also write cosh-1 as 2sin²(h/2)
(by applying the property that cos2x=1-sin²x)
After this multiply the numerator and denominator with h² so that we can apply the property that
[tex]\lim_{x \to \ 0 } sinx/x =1[/tex]
Now your equation will look like this.
[tex]e^{lim_{h \to \ 0}{((cos(h)^2(2sin^2(h/2)*h^2)/(sin(h)^2*h^2)}[/tex]
We will now apply the result
[tex]\lim_{x \to \ 0 } sinx/x =1[/tex]
where x=h²
we get
[tex]e^{lim_{h \to \ 0}{((cos(h)^2(2sin^2(h/2))/(h^2)}[/tex]
we now multiply the numerator and denominator with 4 so that we can say
[tex]\lim_{h^2 \to \ 0 } sin^2(h/2)/(h^2/4) = 1[/tex]
[tex]e^{lim_{h \to \ 0}{((cos(h)^2(2sin^2(h/2))/(h^2*4/4)}[/tex]
[tex]=e^{lim_{h \to \ 0}{((cos(h)^2*2)/(4)}[/tex]
Apply the limits and you will get
[tex]e^{cos(0)^2*2/4[/tex]
[tex]=e^{1/2}[/tex]
Hope this helps.
