Limits

[gmontalv]

I am quite confused on limits

If k(x) = x^3 - 2x -4
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x - 2 find lim k(x)
x->2


Please explain!

 

[soroban]

Hello, gmontalv!

\(\displaystyle \text{If }\,k(x)\:=\:\L\frac{x^3\,-\,2x\,-\,4}{x\,-\,2}\)

$\displaystyle \;\;\;\text{find: }\,\lim_{x\to2}\,k(x)$

Note that: $\displaystyle \,k(2)\,=\,\frac{0}{0}$, an indetermante form.

This means that some cancelling is possible.
$\displaystyle \;\;$ and we suspect that $\displaystyle x\,-\,2$ is a factor of the numerator.


Using long (or synthetic) division, we factor the numerator.

$\displaystyle \;\;\lim_{x\to2}\,\frac{(x\,-\,2)(x^2\,+\,2x\,+\,2)}{x\,-\,2} \;= \;\lim_{x\to2}\,(x^2\,+\,2x\,+\,2) \;= \;2^2\,+\,2\cdot2\,+\,2\;=\;10$