CourseCourseQ BankQuestion BankDocsDocuments

1.7 Selecting Procedures for Determining Limits

Syllabus
2020
Topic
1.7
Level
—

Choosing a Procedure for a Limit

Choose a limit procedure from the information you are given and the form you obtain from the simplest valid check. Start with the least complex method that can justify the result.

What is given or observed? Procedure to try
A graph Trace the output from the left and right and compare the approached values
A table Use inputs increasingly close to the target from both sides and estimate the common trend
An algebraic expression with a defined direct-substitution value Apply substitution and the relevant limit laws, checking their conditions
Direct substitution gives 0/00/0 Rewrite equivalently by factoring, using a conjugate, or applying a suitable identity
A one-sided limit or piecewise rule Use only the branch and direction named, then compare sides only if a two-sided limit is required
  1. Identify the representation and whether the limit is one-sided or two-sided.
  2. Try the simplest applicable check.
  3. Treat a defined value as a possible conclusion, but treat 0/00/0 as a signal to simplify.
  4. Verify all denominator, side, and theorem conditions before stating the limit.

For lim⁡x→2x2−4x−2\displaystyle\lim_{x\to2}\frac{x^2-4}{x-2}, direct substitution produces 0/00/0, so the quotient theorem cannot finish the problem. The shared factor is the diagnostic clue: factor x2−4=(x−2)(x+2)x^2-4=(x-2)(x+2), cancel for x≠2x\ne2, and evaluate lim⁡x→2(x+2)=4\lim_{x\to2}(x+2)=4.

A procedure is justified by the problem's structure, not by preference. Do not force algebra onto a graph, read a two-sided conclusion from only one side, or declare that 0/00/0 means the limit does not exist. Squeeze-theorem selection is developed separately in Topic 1.8.