Exercise · ZW-LIMIT-LAWS-009 · ★☆☆ Basic
When is it enough to substitute \(x=a\) to obtain the limit of a function?
Choose one option.
Show answer
- B. When the function is polynomial or rational with nonzero denominator at \(a\)
Explanation
For polynomial or rational functions with nonzero denominator at \(a\), the limit is the value of the function at \(a\). A \(0/0\) forces you to simplify first.
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-007 Compute lim _x→ 2 (x^2-4)/(x-2).
- ZW-LIMIT-LAWS-008 Compute lim _x→ 1[(x^2-1)/(x-1)+(2)/(x+1)].