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?(a) For \(f(x)=\frac{x}{\ln x}\) find each of the following limits.\(\text { (i) } \lim
Chapter 2, Problem 43(choose chapter or problem)
(a) For \(f(x)=\frac{x}{\ln x}\) find each of the following limits.
\(\text { (i) } \lim _{x \rightarrow 0^{+}} f(x)\)
\(\text { (ii) } \lim _{x \rightarrow 1^{-}} f(x)\)
\(\text { (iii) } \lim _{x \rightarrow 1^{+}} f(x)\)
(b) Use a table of values to estimate \(\lim _{x \rightarrow \infty} f(x)\).
(c) Use the information from parts (a) and (b) to make a rough sketch of the graph of \(f\)
Questions & Answers
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QUESTION:
(a) For \(f(x)=\frac{x}{\ln x}\) find each of the following limits.
\(\text { (i) } \lim _{x \rightarrow 0^{+}} f(x)\)
\(\text { (ii) } \lim _{x \rightarrow 1^{-}} f(x)\)
\(\text { (iii) } \lim _{x \rightarrow 1^{+}} f(x)\)
(b) Use a table of values to estimate \(\lim _{x \rightarrow \infty} f(x)\).
(c) Use the information from parts (a) and (b) to make a rough sketch of the graph of \(f\)
ANSWER:Step 1 of 4
Limit of a Function: In calculus, the limit of any function can be defined as a parameter through which one can analyze the behavior of that particular function limiting it to a particular point within a domain.
(a) The given function is: \(f(x)=\frac{x}{\ln x}\)
Now,
(i) Solving for \(\lim _{x \rightarrow 0^{+}} f(x)\):
\(\begin{aligned} \lim _{x \rightarrow 0^{+}} f(x) & =\lim _{x \rightarrow 0^{+}} \frac{x}{\ln x} \\ & =\frac{\lim _{x \rightarrow 0^{+}} x}{\lim _{x \rightarrow 0^{+}} \ln x} \\ & =\frac{0}{-\infty} \\ & =0 \end{aligned}\)
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