Limits, Continuity & Differentiability
0/0 Limit — Taylor Expansion Coefficient Matching
nta_pyq_2026_jan
Grade None
Question:
If $\displaystyle\lim_{x\to0}\frac{e^{(a-1)x}+2\cos bx+(c-2)e^{-x}}{x\cos z-\log_e(1+z)}=2$, then $a^2+b^2+c^2$ is equal to:
Step-by-Step Solution
Key Concept: Expand numerator and denominator using Taylor series around $x=0$. Denominator $\sim\tfrac{x^2}{2}+\cdots$ For a finite limit of 2, numerator must vanish at $x=0$ (constant term $=0$) and linear term must also $=0$.
$a=-2$, $b^2=2$, $c=-1$. $a^2+b^2+c^2=7$.
Correct Answer: 4