Question: Let f(x + y) = f(x) f(y) & f(x) = 1 + xg(x)h(x), where $\lim_{x \rightarrow 0}$g(x) = 4 and \(\li...

Question

Let f(x + y) = f(x) f(y) & f(x) = 1 + xg(x)h(x), where

$\lim_{x \rightarrow 0}$ g(x) = 4 and $\lim_{x \rightarrow 0}$ h(x) = 2. Then f ¢(x) =

Answer 1

f ¢(x) = $\lim _ { h \rightarrow 0 }$ $\frac{f(x + h) - f(x)}{h}$

= $\lim _ { h \rightarrow 0 }$ $\frac{f(x)f(h) - f(x)}{h}$

= $\lim _ { h \rightarrow 0 }$ $\frac{f(x)(1 + hg(h)\varphi(h) - 1)}{h}$

= $\lim_{h \rightarrow 0}$ f(x). g(h) f(h)

f(x) . 4. 2 = 8 f(x)

Question: Let f(x + y) = f(x) f(y) & f(x) = 1 + xg(x)h(x), where \(\lim_{x \rightarrow 0}\)g(x) = 4 and \(\li...