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Consider the following statements: $\quad[2017-I]$
1- If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exist, then $\lim _{x \rightarrow a}\{f(x) g$
(x) $\}$ exists.
2- If $\lim _{x \rightarrow a}\{f(x) g(x)\}$ exists, then both $\lim _{x \rightarrow a} f(x)$ and
$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ must exist.
Which of the above statements is/are correct?
Options:
1- If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exist, then $\lim _{x \rightarrow a}\{f(x) g$
(x) $\}$ exists.
2- If $\lim _{x \rightarrow a}\{f(x) g(x)\}$ exists, then both $\lim _{x \rightarrow a} f(x)$ and
$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ must exist.
Which of the above statements is/are correct?
Solution:
1424 Upvotes
Verified Answer
The correct answer is:
1 only
If $\lim _{x \rightarrow a} f(x)$ and $\lim _{x \rightarrow a} g(x)$ both exists, then
$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists. But if $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists, then
it is not necesary that both $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ and $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ exists.
$\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists. But if $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})$ exists, then
it is not necesary that both $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ and $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{g}(\mathrm{x})$ exists.
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