Operations of Limits
Always try direct substitution first. This means plug in the ‘c’ value, whatever value x is ap- proaching, directly into the equation. For example:
If your direct substitution results in a numerical answer, you finished! The number you get is your answer- this is what y-value your function is approaching at the given x-value.
Sometimes direct substitution will give you an indeterminate form like:
Sometimes direct substitution will give you an indeterminate form like:
If this happens manipulate your equation until you can use direct substitution. You can do this by factoring, rationalizing, or using special trig limits:
1. Factor, cancel, and try direct substitution again.
1. Factor, cancel, and try direct substitution again.
2. Rationalize, cancel, and try direct substitution again.
3. Special trig limits (manipulate your limit to get one of these two forms):
If direct substitution yields an answer of k/0 , then the limit will either equal positive/ negative infinity, or will not exist. This means you have a vertical asymptote at x=c. To figure out which of these it is, plug in values on either side of ‘c’ into your function and see if you get a positive or negative answer. If the signs do not match, the limit does not exist.