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Question Number 1369 by 123456 last updated on 25/Jul/15

lets f : R \to R continuous and differentiable

compute

g (x) = \lim_{Δ x \to 0} \frac{e^{f (x + Δ x)} - e^{f (x)}}{f (x + Δ x) - f (x)}

Answered by prakash jain last updated on 27/Jul/15

e^{f (x + Δ x)} = 1 + f (x + Δ x) + \frac{f {(x + Δ x)}^{2}}{2!} + \frac{f {(x + Δ x)}^{3}}{3!} +

e^{f (x)} = 1 + f (x) + \frac{f {(x)}^{2}}{2!} + \frac{f {(x)}^{3}}{3!} +

\lim_{Δ x \to 0} \frac{e^{f (x + Δ x)} - e^{f (x)}}{f (x + Δ x) - f (x)} = 1 + \frac{2 f (x)}{2!} + \frac{3 f {(x)}^{2}}{3!} + . .

= e^{f (x)}