Concept preview

What you will be able to do

1. 01Choose an efficient first move for any limit by attempting direct substitution and reading the result as substitution succeeds, the indeterminate form $\tfrac{0}{0}$, or a nonzero-over-zero form.
2. 02Match each outcome to the correct follow-up procedure: accept the value, apply algebra, read a given graph or table, or diagnose that the limit does not exist.
3. 03Distinguish an existing limit from one that does not exist by recognizing jumps, unequal one-sided limits, and vertical asymptotes.

A strong calculus student is not someone who knows more tricks than everyone else — it is someone who selects the right tool immediately. Every limit you meet fits a small number of cases, and each case has a designated response. Learning to classify the limit at a glance is what makes the rest fast and accurate.