2021-11-18

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval.

Othrom

f(x) is continous at x=a if and only if
$\underset{x\to a}{lim}f\left(x\right)=f\left(a\right)$
$\underset{x\to a}{lim}f\left(x\right)=\underset{x\to a}{lim}\frac{2x+3}{x-2}$
Divisionlaw for lomits
$\underset{x\to a}{lim}\frac{2x+3}{x-2}=\frac{\underset{x\to a}{lim}2x+3}{\underset{x\to a}{lim}x-2}$
$\frac{\underset{x\to a}{lim}2x+3}{\underset{x\to a}{lim}x-2}=\frac{\underset{x\to a}{lim}2x+\underset{x\to a}{lim}3}{\underset{x\to a}{lim}x-2}$
$\frac{\underset{x\to a}{lim}2x+\underset{x\to a}{lim}3}{\underset{x\to a}{lim}x-\underset{x\to a}{lim}2}=\frac{2\cdot \underset{x\to a}{lim}x+\underset{x\to a}{lim}3}{\underset{x\to a}{lim}x-\underset{x\to a}{lim}2}$
$\frac{2\cdot \underset{x\to a}{lim}x+\underset{x\to a}{lim}3}{\underset{x\to a}{lim}x-\underset{x\to a}{lim}2}=\frac{2a+3}{a-2}$
$\frac{2a+3}{a-2}=f\left(a\right)$
Hence proved that f(x) is continuous for all $a\ne 2$

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