Problem 1b Explanation

Problem 1: Evaluate the following limits. When appropriate, declare whether the limit goes to ∞ or -∞ if the limit does not exist.



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!

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