A reliable method
- Check whether direct substitution is legitimate.
- If an indeterminate form appears, simplify for nearby inputs.
- Compare left and right limiting behavior.
- For continuity compare that limit with the function value.
Example 1
Find \(\lim_{x\to 3}(\frac{x^{2} - 9}{x - 3})\).
- Try the limiting input, but distinguish an undefined original value from a nonexistent limit.
- For a removable rational expression, factor and cancel only for nearby inputs where the canceled factor is nonzero: \(x + 3\).
- The left and right limits agree at \(6\). The limit describes nearby behavior, not necessarily the value at the point.
Answer: \(6\)
Example 2
Find \(\lim_{x\to 1}(\frac{x^{2} - 1}{x - 1})\).
- Try the limiting input, but distinguish an undefined original value from a nonexistent limit.
- For a removable rational expression, factor and cancel only for nearby inputs where the canceled factor is nonzero: \(x + 1\).
- The left and right limits agree at \(2\). The limit describes nearby behavior, not necessarily the value at the point.
Answer: \(2\)
Common mistakes
- An undefined value at a point does not automatically mean the limit fails.
- A two-sided limit requires agreement from both directions.
Practice without the answer first
- Find \(\lim_{x\to 2}(2 x^{2} + 3)\).
- Find \(\lim_{x\to 0}(\frac{x^{2} + 5 x}{x})\).
- Find \(\lim_{x\to 2}(\frac{x^{3} - 8}{x - 2})\).
Answers and step-by-step explanations
Practice 1
- Try the limiting input, but distinguish an undefined original value from a nonexistent limit.
- For a removable rational expression, factor and cancel only for nearby inputs where the canceled factor is nonzero: \(2 x^{2} + 3\).
- The left and right limits agree at \(11\). The limit describes nearby behavior, not necessarily the value at the point.
Answer: \(11\)
Practice 2
- Try the limiting input, but distinguish an undefined original value from a nonexistent limit.
- For a removable rational expression, factor and cancel only for nearby inputs where the canceled factor is nonzero: \(x + 5\).
- The left and right limits agree at \(5\). The limit describes nearby behavior, not necessarily the value at the point.
Answer: \(5\)
Practice 3
- Try the limiting input, but distinguish an undefined original value from a nonexistent limit.
- For a removable rational expression, factor and cancel only for nearby inputs where the canceled factor is nonzero: \(x^{2} + 2 x + 4\).
- The left and right limits agree at \(12\). The limit describes nearby behavior, not necessarily the value at the point.
Answer: \(12\)
These original skill exercises are not official exam questions or a score prediction. Use the coverage notes on your exam hub to choose relevant practice.
