Solving limits with factoring
When substitution doesn’t solve the limit, try factoring
When you can’t just plug in the value you’re evaluating, your next approach should be factoring.
Factor the function to cancel any removable discontinuities, then evaluate the limit
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How to evaluate a limit using factoring and cancellation
Example
Evaluate the limit.
???\lim_{x\to 4}\frac{x^2-16}{x-4}???
Just plugging in ???4??? would give us a nasty ???0/0??? result. Therefore, we’ll try factoring instead.
???\lim_{x\to 4}\frac{(x+4)(x-4)}{x-4}???
Canceling ???(x-4)??? from the top and bottom of the fraction leaves us with something that is much easier to evaluate:
???\lim_{x\to 4}x+4???
Now the problem is simple enough that we can use substitution to find the limit.
???4+4???
???8???