Limits, Continuity & Differentiability
L'Hôpital — Cube Root and Square Root Expressions
nta_pyq_2024_apr
Grade 12
Question:
If $\displaystyle\lim_{x\to1}\frac{(5x+1)^{1/3}-(x+5)^{1/3}}{(2x+3)^{1/2}-(x+4)^{1/2}}=\frac{m\sqrt{5}}{n(2n)^{2/3}}$, where $\gcd(m,n)=1$, then $8m+12n$ is equal to
Step-by-Step Solution
Key Concept: Apply L'Hôpital's rule (0/0 form at $x=1$). Differentiate numerator and denominator separately, then evaluate at $x=1$.
$m=8$, $n=3$. $8m+12n=100$.
Correct Answer: 100