Consider the following statements:

1. If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) both exist, then \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists.

2. If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists, then both \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) must exist.

Which of the above statements is/are correct?

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  1. 1 only
  2. 2 only
  3. Both 1 and 2
  4. Neither 1 nor 2

Answer (Detailed Solution Below)

Option 1 : 1 only
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Detailed Solution

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Concept:

Properties of Limits:

  • If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) both exist, then \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists.
  • If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists, then it’s not necessary \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) both exists.

 

Calculation:

We know that, if \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) both exist, then \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists.

Hence, statement (1) is correct.

We know that, If \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} \left\{ {{\rm{f}}\left( {\rm{x}} \right){\rm{g}}\left( {\rm{x}} \right)} \right\}\) exists, then it’s not necessary \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{f}}\left( {\rm{x}} \right)\) and \(\mathop {\lim }\limits_{{\rm{x}} \to {\rm{a}}} {\rm{g}}\left( {\rm{x}} \right)\) both exists.

Hence, statement (2) is incorrect.

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