The equivalent limit of the function as x tends to zero is -1/2
Given the limit of a function:
[tex]f(x)=\frac{cosx-1}{sin2x}[/tex]
Taking the limit of the function as x tends to zero
[tex]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)[/tex]
Applying the L'hospital rule will give:
[tex]\lim_{x \to 0} \frac{-sinx}{2sinxcosx}\\= \frac{-sin0}{2sin0cos0}\\=\frac{0}{0}(ind)[/tex]
Applying the l'hospital rule the second time:
[tex]\lim_{x \to 0} \frac{-cosx}{2(-sin^2x+cos^2x)}\\= \frac{-cos0}{2(-sin^20+cos^20)}\\=\frac{-1}{2}[/tex]
Hence the equivalent limit of the function as x tends to zero is -1/2
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