Category 1 is complete, Category 2 is the one I need help with, using Category 1

Question

Category 1 is complete, Category 2 is the one I need help with, using Category 1 as reference. Thank you.

Answer to Category 1:

Rule discovered:

If the value of f(x) at the limiting value of x exists, then we can find it by directly putting x in the expression for f(x).

The problem in is that at x=5, denominator becomes 0, thus f(x) becomes infinity, so limit doesn't exist.

Category 2:

What process can you develop that can be used to solve any category two problem?   What issues

arise when you try to use the rule developed for category one problems? If you were to graph these functions what feature might exist in the graph at the limiting value? Why?

Explanation / Answer

We can try to factorize both numerator and denominator and cancel out the term which makes the denominator zero.

For example in first case, we can factorize (x^2 - 9) = (x+3)(x-3), and cancel (x-3) at denominator and get (x+3) = 6.

The problem with category 1 problem: The fraction is becoming 0/0 which is indeterminate form, so we can't directy substitute.

Graph: The exact value at that point will not exist, but the right ahnd limit and left liimit will exist and will be same

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