Limits, Continuity & Differentiability
Continuity And Differentiability
nta_abhyas_2025
Grade 12
Question:
If $f(x) = \begin{cases} \frac{x \ln(e + \cos x)}{x^2} & x > 0 \\ q & x \leq 0 \end{cases}$ is continuous at $x = 0$, then the value of $\frac{pq+1}{q}$ is
Step-by-Step Solution
Key Concept: Use continuity conditions at a point and L'Hôpital's rule to establish relationships between parameters in piecewise functions.
For continuity at $x = 0$: $\lim_{x \to 0} f(x) = f(0)$ requires $a(\frac{c}{a} + \frac{c}{b}) + c(\frac{c}{a} + \frac{c}{b}) = q$. Simplifying the limit expression with L'Hôpital's rule and comparing coefficients: $a + b = 0$, $c = 0$, and $(- \frac{c}{a}) = q$ gives us $c = 0, a = -18, b = 18$.
Correct Answer: 18