Limits, Continuity & Differentiability
Indeterminate Limit — Taylor Expansion
nta_pyq_2023_apr
Grade 12
Question:
If $\displaystyle\lim_{x\to0}\dfrac{e^{ax}-\cos(bx)-cxe^{-cx}}{1-\cos(2x)}=17$, then $5a^2+b^2$ is equal to
Step-by-Step Solution
Key Concept: Expand numerator: $(a-\frac{c}{2})x+(\frac{a^2+b^2+c^2}{2})x^2+\cdots$. Denominator $\sim2x^2$. For limit to exist: $a-\frac{c}{2}=0\Rightarrow c=2a$.
$c=2a$. $5a^2+b^2=68$.
Correct Answer: 3