Consider $f(x) = x \ln x$ and $g(x) = e^{2x}$. Let $a$ and $b$ be two values of $x$ satisfying $f(x) = g(x)$ with $a If \(\lim_{x \to b} \frac{f(x) - c}{g(x) - b^2} = l$, then the value of $c - l$ equals:

Limits, Continuity & Differentiability

Limit Computation

Grade 12

Question:

Consider $f(x) = x \ln x$ and $g(x) = e^{2x}$. Let $a$ and $b$ be two values of $x$ satisfying $f(x) = g(x)$ with $a < b$.If $\lim_{x \to b} \frac{f(x) - c}{g(x) - b^2} = l$, then the value of $c - l$ equals:

(a) $4 - e^2$

(b) $e^2 - 4$

(c) $4 - e$

(d) $e - 4$

Step-by-Step Solution

Key Concept: Use L'Hôpital's rule or Taylor expansion around the point $x = b$ where $f(b) = g(b)$. The condition $f(b) = g(b)$ means $b\ln b = e^{2b}$.

Answer: (b)

Correct Answer: B

Question:

Step-by-Step Solution

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