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 $\underset{x\to 0}{lim}f\left(x\right)=\frac{1}{2}$ and $\underset{x\to 0}{lim}g\left(x\right)=\sqrt{2}$. Find the limits as $x\to 0$ of the following functions. $f\left(x\right)\frac{\mathrm{cos}x}{x-1}$
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Given
$\underset{x\to 0}{lim}f\left(x\right)=\frac{1}{2}$ and $limg\left(x\right)=\sqrt{2}$
we have to find
$f\left(x\right)\frac{\mathrm{cos}x}{x-1}$
we know
$\underset{x\to 0}{lim}f\left(x\right)\cdot g\left(x\right)=\left[\underset{x\to 0}{lim}f\left(x\right)\right]\left[\underset{x\to 0}{lim}g\left(x\right)\right]$
and $\underset{x\to 0}{lim}\frac{f\left(x\right)}{g\left(x\right)}=\underset{x\to 0}{lim}\frac{f\left(x\right)}{\underset{x\to 0}{lim}g\left(x\right)}$
$\underset{x\to 0}{lim}\frac{f\left(x\right)\mathrm{cos}x}{x-1}=\left[\underset{x\to 0}{lim}f\left(x\right)\right]\cdot \left[\underset{x\to 0}{lim}\frac{\mathrm{cos}x}{x-1}\right]$
$=\frac{\frac{1}{2}\underset{x\to 0}{lim}\mathrm{cos}x}{\underset{x\to 0}{lim}\left(x-1\right)}$
$=\frac{1}{2}\cdot \frac{1}{-1}$
$=-\frac{1}{2}$