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  4. Let f(x)=(x2-9 x+20/x-[x])( where [x] is the greatest integer not greater than x ), then

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Q. Let $f(x)=\frac{x^{2}-9 x+20}{x-[x]}($ where $[x]$ is the greatest integer not greater than $x$ ), then

Limits and Derivatives

Solution:

$=\displaystyle\lim _{x \rightarrow 5^{-}} \frac{x^{2}-9 x+20}{x-[x]}$
$=\displaystyle\lim _{x \rightarrow 5^{-}} \frac{(x-5)(x-4)}{x-4}$
$=\displaystyle\lim _{x \rightarrow 5^{-}}(x-5)=0$
$=\displaystyle\lim _{x \rightarrow 5^{-}} \frac{x^{2}-9 x+20}{x-[x]}$
$=\displaystyle\lim _{x \rightarrow 5^{+}} \frac{(x-5)(x-4)}{x-5}$
$=\displaystyle\lim _{x \rightarrow 5^{+}}(x-4)=1$