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Which of the following is true for the real valued functions f(x),h(x),g(x) 

\lim _{x \rightarrow a} f(x)=L, \lim _{x \rightarrow-a} g(x)=L, \lim _{x \rightarrow-a} f(x)=N, \lim _{x \rightarrow a} g(x)=N?

Option: 1

Sandwich theorems can be applied to find \lim _{x \rightarrow a} h(x) \lim _{x \rightarrow a} f(x) \text { and } \lim _{x \rightarrow a} g(x) exists.


Option: 2

Sandwich theorems can be applied to find \lim _{x \rightarrow a} h(x) \lim _{x \rightarrow a} f(x) \text { and } \lim _{x \rightarrow a} h(x) exists.


Option: 3

Both a and b


Option: 4

None of the above


Answers (1)

best_answer

Note the following points.

  • The limit of any function \lim _{x \rightarrow p} f(x) may exist even when f(x) is not defined at x=p 

  • The application of the Sandwich theorem needs the real valued functions to share the same domain of consideration.

  • The application of the Sandwich theorem needs the limits \lim _{x \rightarrow a} f(x) and \lim _{x \rightarrow a} g(x) or \lim _{x \rightarrow a} f(x) and \lim _{x \rightarrow a} g(x) .

  • It is true from the following that the limits limits \lim _{x \rightarrow a} f(x) and \lim _{x \rightarrow a} g(x) or \lim _{x \rightarrow a} f(x) and \lim _{x \rightarrow a} g(x)  exists.

But the domains of the real valued functions f(x),h(x), g(x) 

 must be defined and same for the application of the Sandwich theorem in the said limits.

Posted by

rishi.raj

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