Problem 1: Evaluate the following limits. When appropriate, declare whether the limit goes to ∞ or -∞ if the limit does not exist.
Answer
Explanation
Remember: always start out with direct substitution! In this case, we're evaluating the limit as x approaches 8...
...therefore, we'll plug in 8 for x...
...and solve...
...resulting in an indeterminate form. Dang it!
Since we got an indeterminate form, then we've gotta try factoring out some terms here. Looking at this problem, I notice that we have polynomials in our numerator and denominator that can be factored. (For complete clarity, I've highlighted these polynomials below.)
Let’s try factoring the numerator into (x - 8)(x + 3)...
...and denominator into (x - 8)(x + 8).
Now, do we have any duplicate terms in the numerator and denominator that can be canceled out?
Yes, (x - 8)!
When we cancel this term out on the numerator and denominator, we get the following:
Now that we've factored out (x - 8), let's try direct substitution again. Plug in x = 8...
...and solve.
...resulting in a value other than 0/0! We have our answer!
As our function approaches x = 8 from the right and left side, it approaches a y-value of 88/16!