3. $limlimits_{x \to 1}\frac{x}{ln x}$ is\n(a) 0\n(b) $\frac{1}{e}$\n(c) 1\n(d) $e$\n(e) nonexistent
3. $limlimits_{x \to 1}\frac{x}{ln x}$ is\n(a) 0\n(b) $\frac{1}{e}$\n(c) 1\n(d) $e$\n(e) nonexistent
Answer
Explanation:
Step1: Check the form
When (x = 1), (\frac{x}{\ln x}=\frac{1}{\ln1}=\frac{1}{0}). As (x\rightarrow1^{+}), (\ln x\rightarrow0^{+}) and (\frac{x}{\ln x}\rightarrow+\infty). As (x\rightarrow1^{-}), (\ln x\rightarrow0^{-}) and (\frac{x}{\ln x}\rightarrow-\infty).
Step2: Determine the limit
Since the left - hand limit (\lim_{x\rightarrow1^{-}}\frac{x}{\ln x}) and the right - hand limit (\lim_{x\rightarrow1^{+}}\frac{x}{\ln x}) are not equal.
Answer:
E. nonexistent