Question

If f(x) = Tan(x)-1 and g(x) = Sin(x) – Cos(x) than find value of limx → ^π⁄4⁡ ^f(x)⁄g(x)

(a) -√2

(b) √(-2)

(c) √2

(d) √3

I had been asked this question in a national level competition.

This is a very interesting question from Indeterminate Forms in portion Differential Calculus of Engineering Mathematics

Answer 1

The correct answer is (c) √2

The best I can explain: \(\lim_{x\rightarrow \pi/4}\frac{f(x)}{g(x)}=\lim_{x\rightarrow \pi/4}\frac{tan(x)-1}{sin(x)-cos(x)}=\frac{(1-1)}{\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}}=0/0\)

Hence this gives indeterminate forms hence by L’Hospital Rule

\(\lim_{x\rightarrow \pi/4}\frac{f(x)}{g(x)}=\lim_{x\rightarrow \pi/4}\frac{f'(x)}{g'(x)}=\lim_{x\rightarrow \pi/4}\frac{sec^2}{cos(x)+sin(x)}=\sqrt{2}\)

Hence

\(\lim_{x\rightarrow \pi/4}\frac{f(x)}{g(x)}=\sqrt{2}\)