COMEDK · Maths · 16. Limits
\(\lim _\limits{x \rightarrow 0} \dfrac{a^x-b^x}{c^x-d^x}=\)
- A \(\infty\)
- B \(\dfrac{\log \left(\dfrac{a}{b}\right)}{\log \left(\dfrac{c}{d}\right)}\)
- C \(\dfrac{\log a b}{\log c d}\)
- D 0
Answer & Solution
Correct Answer
(B) \(\dfrac{\log \left(\dfrac{a}{b}\right)}{\log \left(\dfrac{c}{d}\right)}\)
Step-by-step Solution
Detailed explanation
The given limit is \(L = \lim_{x \rightarrow 0} \dfrac{a^x - b^x}{c^x - d^x}\). Since the limit is of the form \(\dfrac{0}{0}\) as \(x \rightarrow 0\), we apply L'Hopital's rule by differentiating the numerator and the denominator with respect to \(x\). The derivative of \(a^x\)…
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