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1.B. Evaluating Limits Algebraically

Another method of finding limits involves finding them algebraically.

Direct substitution can be used for expressions that are continuous at the value that x is getting closer to.

The dividing out technique and rationalizing technique can be used to evaluate limits in which direct substitution yields an indeterminate.

Besides the techniques listed above, some rules to help us find limits are listed below (provided limits exist):

Sum of a limit equals limit of their sum:
Difference of a limit equals limit of their difference:
Product of a limit equals limit of their product:
Quotient of a limit equals limit of the quotient (limit of denominator does not equal zero):
Composite of a limit equals limit of the composite (provided limits exist and f is continuous at g's limit):
Infinite Limits involves limits that look like the following:
Limits at Infinity involves limits that look like the following:
Essential Questions
  • How are limits evaluated analytically?
  • How can direct substitution, dividing out technique, and rationalizing technique used to evaluate limits?
Practice Problems