Question

L’Hospital Rule states that

(a) If \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}\) is an indeterminate form than \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}\) if \(\lim_{x\rightarrow a} \frac{f'(x)}{g'(x)}\) has a finite value

(b) \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}\) always equals to \(\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}\)

(c) \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}\) if an indeterminate form than cannot be solved

(d) \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}\) if an indeterminate form than it is equals to zero.

I got this question in exam.

I'd like to ask this question from Indeterminate Forms topic in division Differential Calculus of Engineering Mathematics

Answer 1

Right choice is (a) If \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}\) is an indeterminate form than \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}\) if \(\lim_{x\rightarrow a} \frac{f'(x)}{g'(x)}\) has a finite value

Explanation: According to L’Hospital Rule, if \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)}\) is indeterminate and \(\lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}\) has a finite value then \(\lim_{x\rightarrow a}\frac{f(x)}{g(x)} = \lim_{x\rightarrow a}\frac{f'(x)}{g'(x)}\). It is helpful in solving limits of indeterminate forms.