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Question

Find each limit. limx3(x2x33xx3)\lim _{x \rightarrow 3}\left(\frac{x^{2}}{x-3}-\frac{3 x}{x-3}\right)

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We would like to find the limit for the following expression.

limx3 (x2x33xx3)\color{#4257b2}\lim\limits_{x\to3}\ \left(\dfrac{x^2}{x-3}-\dfrac{3x}{x-3}\right)

\lozenge\ \ Solution:\text{\underline{\bf{Solution:}}}

Simplify the expression as follows:

limx3 (x2x33xx3)=limx3 (x23xx3)\lim\limits_{x\to3}\ \left(\dfrac{x^2}{x-3}-\dfrac{3x}{x-3}\right)=\lim\limits_{x\to3}\ \left(\dfrac{x^2-3x}{x-3}\right)

Use GCF property as follows:

limx3 (x(x3)x3)\lim\limits_{x\to3}\ \left(\dfrac{x(x-3)}{x-3}\right)

limx3 (x)\lim\limits_{x\to3}\ (x)

According the rule of the limit property as follows:

  limxa (x)=a\because\ \ \lim\limits_{x\to a}\ (x)=a

  limx3 (x)=3\therefore\ \ \lim\limits_{x\to3}\ (x)=3

limx3 (x2x33xx3)=3\color{#c34632}\lim\limits_{x\to3}\ \left(\dfrac{x^2}{x-3}-\dfrac{3x}{x-3}\right)=3

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