# 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
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. $$f(x)\frac{\cos x}{x-1}$$

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 $$\lim_{x\rightarrow0}\frac{f(x)}{g(x)}=\lim_{x\rightarrow0} \frac{f(x)}{\lim_{x\rightarrow0}g(x)}$$
$$\lim_{x\rightarrow0}\frac{f(x)\cos x}{x-1}=[\lim_{x\rightarrow0} f(x)]\cdot[\lim_{x\rightarrow0}\frac{\cos x}{x-1}]$$
$$=\frac{\frac{1}{2}\lim_{x\rightarrow0}\cos x}{\lim_{x\rightarrow0}(x-1)}$$
$$=\frac{1}{2}\cdot\frac{1}{-1}$$
$$=-\frac{1}{2}$$

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