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(Video) Calculating a Limit by Factoring and Canceling

by PatrickJMT

Here's a video by patrickJMT showing you how to find a limit of a function using the factor canceling technique.

He demonstrates this technique by looking at an example: $$\lim_{x\rightarrow2}\frac{x^2-4}{x-2}$$ Notice what happens when we plug in 2 straight away. $$\begin{align} \frac{x^2-4}{x-2}&=\frac{\color{red}{(2)}^2-4}{\color{red}{(2)}-2}\\ &=\frac{4-4}{2-2}\\&=\frac00 \end{align}$$ So we have to be creative and figure something else out. We can instead factor this expression. $$\lim_{x\rightarrow2}\frac{x^2-4}{x-2}=\lim_{x\rightarrow2}\frac{(x+2)(x-2)}{x-2}$$ When we factor the top we get an $x-2$ in both the numerator and denominator. This can be canceled out. $$\lim_{x\rightarrow2}\frac{(x+2)\color{red}{(x-2)}}{\color{red}{x-2}}=\lim_{x\rightarrow2}x+2$$ Now that it's simplified let's try plugging 2 back in. $$x+2=\color{red}{(2)}+2=4$$ So $$\lim_{x\rightarrow2}\frac{x^2-4}{x-2}=4$$

If we graph the function $y=\dfrac{x^2-4}{x-2}$ it's going to look a lot like the function $y=x+2$ except it will be undefined at $x=2$ because we have a zero in the denominator.

As you can see when we put in points getting closer and closer to 2, the function gets closer and closer to 4.