# Find the limit. lim_{xrightarrow(pi/2)}frac{sec x}{tan x}

Find the limit.
$\underset{x\to \left(\pi /2\right)}{lim}\frac{\mathrm{sec}x}{\mathrm{tan}x}$
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We have to evaluate
$\underset{x\to \left(\pi /2{\right)}^{-}}{lim}\frac{\mathrm{sec}x}{\mathrm{tan}x}$
We can see that if we put given limit then we get $\frac{\mathrm{\infty }}{\mathrm{\infty }}$ form.
So, to remove this form, we will simplify the given expression and then we will put limit.
We know that $\mathrm{sec}x=\frac{1}{\mathrm{cos}x}$ and $\mathrm{tan}x=\frac{\mathrm{sin}x}{\mathrm{cos}x}$
Therefore,
$\underset{x\to \left(\pi /2{\right)}^{-}}{lim}\frac{\mathrm{sec}x}{\mathrm{tan}x}=\underset{x\to \left(\pi /2{\right)}^{-}}{lim}\frac{\frac{1}{\mathrm{cos}x}}{\frac{\mathrm{sin}x}{\mathrm{cos}x}}$
$\underset{x\to \pi /2}{lim}\frac{1}{\mathrm{sin}x}$
$=\frac{1}{\mathrm{sin}\frac{\pi }{2}}$
$=\frac{1}{1}$
$=1$
Hence, required answer is 1 .
Jeffrey Jordon