Question

Limif f(x)=cosx−1sin2x, then limx→0f(x) is equivalent to which of the following?

Answer 1

The equivalent limit of the function as x tends to zero is -1/2

Given the limit of a function:

$f(x)=\frac{cosx-1}{sin2x}$

Taking the limit of the function as x tends to zero

$f(x)=\frac{cosx-1}{sin2x}\\ \lim_{x\to 0} \frac{cosx-1}{sin2x}\\=\frac{cos0-1}{sin2(0)}\\=\frac{1-1}{0}\\=\frac{0}{0}(ind)$

Applying the L'hospital rule will give:

$\lim_{x \to 0} \frac{-sinx}{2sinxcosx}\\= \frac{-sin0}{2sin0cos0}\\=\frac{0}{0}(ind)$

Applying the l'hospital rule the second time:

$\lim_{x \to 0} \frac{-cosx}{2(-sin^2x+cos^2x)}\\= \frac{-cos0}{2(-sin^20+cos^20)}\\=\frac{-1}{2}$

Hence the equivalent limit of the function as x tends to zero is -1/2

Learn more here: brainly.com/question/20512128