Introduction to the concept of limits - Worksheet

Evaluate the limit of a rational function as x approaches a finite value using direct substitution

Factor and simplify a 0/0 indeterminate form to find a limit at a removable discontinuity

Determine when a limit is infinite or does not exist (DNE) from a rational function's behavior near the target value

Group the problems by technique before assigning: problems 1-13 mostly require factor-and-cancel simplification, while problems 14-18 test recognizing infinite or nonexistent limits — useful for splitting a lesson in two.

Have students attempt direct substitution first on every problem, since several (e.g. problem 1) look like they will produce a 0/0 form until factored, which mirrors the worksheet's core skill.

Use the answer key's DNE and infinity notations as a discussion point — ask students to explain in words why a given limit is undefined rather than just checking the final numeral.

Direct substitution — students plug the target x-value directly into expressions that are already defined there, such as problem 12's (3x²−24x+36)/(x−6).

Factoring rational expressions to remove a removable discontinuity — problem 1's (x²−25)/(x−5) simplifies by canceling the (x−5) factor before substituting.

Recognizing infinite limits — problem 17's 1/(x−3)² grows without bound as the denominator approaches zero from both sides.

Identifying limits that do not exist (DNE) — problems 14, 15, 16, and 18 require recognizing when a one-sided mismatch or a zero denominator makes the limit undefined.