Exercise · ZW-INDETERMINATE-FORMS-007 · ★★☆ Intermediate
Compute \(\lim_{x\to 0}\,\frac{\sqrt{x + 1} - 1}{x}\).
Choose one option.
Show answer
- D. \(\frac{1}{2}\)
Explanation
Substituting x = 0 gives 0/0. Multiply numerator and denominator by the conjugate \(\sqrt{x+1}+1\): \(\dfrac{x}{x(\sqrt{x+1}+1)}=\dfrac{1}{\sqrt{x+1}+1}\), and substituting x = 0 leaves \(\frac{1}{2}\).
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-004 Compute lim _x→ 2 (x^3 - 8)/(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-008 Which of these expressions is an indeterminate form?
- ZW-INDETERMINATE-FORMS-009 Compute lim _x→ 0 (√x + 4 - 2)/(x).