Exercise · ZW-INDETERMINATE-FORMS-004 · ★★☆ Intermediate
Compute \(\lim_{x\to 2}\,\frac{x^{3} - 8}{x - 2}\).
Choose one option.
Show answer
- C. \(12\)
Explanation
Substituting x = 2 gives 0/0. Factor: \(\dfrac{\left(x - 2\right) \left(x^{2} + 2 x + 4\right)}{x - 2}\). Cancelling the common factor leaves \(x^{2} + 2 x + 4\), and substituting x = 2 gives \(12\).
Review the concept: Indeterminate forms
An indeterminate form is a limit expression, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), whose value cannot be deduced from the limits of its parts.
\[\frac{0}{0},\ \frac{\infty}{\infty},\ 0\cdot\infty,\ \infty-\infty\]
More practice problems: indeterminate forms
- ZW-INDETERMINATE-FORMS-001 Which of these limits has the indeterminate form 0/0 on direct substitution?
- ZW-INDETERMINATE-FORMS-002 Compute lim _x→ 3 (x^2 - 9)/(x - 3).
- ZW-INDETERMINATE-FORMS-003 Compute lim _x→ 2 (x^2 - 5 x + 6)/(x - 2).
- ZW-INDETERMINATE-FORMS-005 Compute lim _x→ 2 (x^2 + x - 6)/(x^2 - 4).
- ZW-INDETERMINATE-FORMS-006 Compute lim _x→ 1 (x^2 - 1)/(x^2 - 3 x + 2).
- ZW-INDETERMINATE-FORMS-007 Compute lim _x→ 0 (√x + 1 - 1)/(x).
- ZW-INDETERMINATE-FORMS-008 Which of these expressions is an indeterminate form?
- ZW-INDETERMINATE-FORMS-009 Compute lim _x→ 0 (√x + 4 - 2)/(x).