Suppose the functions f(x) and g(x) are defined for all x and that lim_{xrightarrow0}f(x)=frac{1}{2} and lim_{xrightarrow0}g(x)=sqrt2. Find the limits as xrightarrow0 of the following functions. f(x)frac{cos x}{x-1}

Suppose the functions f(x) and g(x) are defined for all x and that lim_{xrightarrow0}f(x)=frac{1}{2} and lim_{xrightarrow0}g(x)=sqrt2. Find the limits as xrightarrow0 of the following functions. f(x)frac{cos x}{x-1}

Question
Limits and continuity
asked 2021-01-15
Suppose the functions f(x) and g(x) are defined for all x and that \(\lim_{x\rightarrow0}f(x)=\frac{1}{2}\) and \(\lim_{x\rightarrow0}g(x)=\sqrt2\). Find the limits as \(x\rightarrow0\) of the following functions. \(f(x)\frac{\cos x}{x-1}\)

Answers (1)

2021-01-16
Given
\(\lim_{x\rightarrow0}f(x)=\frac{1}{2}\) and \(\lim g(x)=\sqrt2\)
we have to find
\(f(x)\frac{\cos x}{x-1}\)
we know
\(\lim_{x\rightarrow0}f(x)\cdot g(x)=[\lim_{x\rightarrow0}f(x)][\lim_{x\rightarrow0}g(x)]\)
and \(\lim_{x\rightarrow0}\frac{f(x)}{g(x)}=\lim_{x\rightarrow0} \frac{f(x)}{\lim_{x\rightarrow0}g(x)}\)
\(\lim_{x\rightarrow0}\frac{f(x)\cos x}{x-1}=[\lim_{x\rightarrow0} f(x)]\cdot[\lim_{x\rightarrow0}\frac{\cos x}{x-1}]\)
\(=\frac{\frac{1}{2}\lim_{x\rightarrow0}\cos x}{\lim_{x\rightarrow0}(x-1)}\)
\(=\frac{1}{2}\cdot\frac{1}{-1}\)
\(=-\frac{1}{2}\)
0

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