Exercise · ZW-LIMIT-LAWS-007 · ★★☆ Intermediate
Compute \(\lim_{x\to 2}\,\frac{x^{2}-4}{x-2}\).
Choose one option.
Show answer
- D. \(4\)
Explanation
Substituting gives \(0/0\), so factor \(x^{2}-4=(x-2)(x+2)\), cancel \(x-2\) for \(x\neq 2\), leaving \(x+2\), which gives 4 at 2.
Review the concept: Limit laws
If \(\lim_{x\to a} f(x)\) and \(\lim_{x\to a} g(x)\) exist, the limit of their sum, difference, product or quotient (with nonzero denominator) is the same operation applied to the limits.
\[\lim_{x\to a}[f(x)\,g(x)]=\lim_{x\to a}f(x)\cdot\lim_{x\to a}g(x)\]
More practice problems: limit laws
- ZW-LIMIT-LAWS-001 Compute lim _x→ 2 (x^2 + 3 x - 1).
- ZW-LIMIT-LAWS-002 Compute lim _x→ 4 (x - 3) (x + 1).
- ZW-LIMIT-LAWS-003 Compute lim _x→ 3 (x^2 + 1)/(x - 1).
- ZW-LIMIT-LAWS-004 Compute lim _x→ 4 √x (x - 1).
- ZW-LIMIT-LAWS-005 By the product law, what does lim _x→ a[f(x) g(x)] equal if both limits exist?
- ZW-LIMIT-LAWS-006 Besides the existence of the limits of numerator and denominator, what does the quotient law requir…
- ZW-LIMIT-LAWS-008 Compute lim _x→ 1[(x^2-1)/(x-1)+(2)/(x+1)].
- ZW-LIMIT-LAWS-009 When is it enough to substitute x=a to obtain the limit of a function?