Let f(x)= displaystyle lim h arrow 0 (( sin (x+h)) ln (x+h)-( sin x) ln x/h), then f((π/2)) is

Question

Let f(x)= displaystyle lim h arrow 0 (( sin (x+h)) ln (x+h)-( sin x) ln x/h), then f((π/2)) is (A) equal to 0 (B) equal to 1 (C)  ln (π/2) (D) non-existent

Tardigrade Admin · Accepted Answer

Let g(x)=( sin x) ln x=e ln x ⋅ ln ( sin x) f(x)=g prime(x)=( sin x) ln x[ cot x( ln x)+( ln ( sin x)/x)] Hence, f ((π/2))= g prime((π/2))=1(0+0)=0

Q. Let $f (x) = h \to 0 lim \frac{( sin ( x + h ) ) ^{l n (x + h)} - ( sin x ) ^{l n x}}{h}$ , then $f (\frac{π}{2})$ is

equal to 0

equal to 1

$ln \frac{π}{2}$

non-existent

Let $g (x) = (sin x)^{l n x} = e^{l n x \cdot l n (s i n x)}$
$f (x) = g^{'} (x) = (sin x)^{l n x} [cot x (ln x) + \frac{l n ( s i n x )}{x}]$
Hence, $f (\frac{π}{2}) = g^{'} (\frac{π}{2}) = 1 (0 + 0) = 0$

Q. Let f(x)=h→0lim​h(sin(x+h))ln(x+h)−(sinx)lnx​, then f(2π​) is