Question

if \\( \lim _{x \
ightarrow a^{-}} f(x)=l \\) and \\( \lim _{x \
ightarrow a^{+}} f(x)=m \\), where \\( l \\) and \\( m \\) are finite real numbers, then how are \\( l \\) and \\( m \\) related if \\( \lim _{x \
ightarrow a} f(x) \\) exists?

choose the correct answer.

\\( \bigcirc \\) a. \\( l = m \\)
\\( \bigcirc \\) b. \\( l \
eq m \\)
\\( \bigcirc \\) c. \\( l < m \\)
\\( \bigcirc \\) d. \\( l > m \\)

Explanation:

Step1: Recall the limit existence condition

The limit \(\lim_{x
ightarrow a}f(x)\) exists if and only if the left - hand limit \(\lim_{x
ightarrow a^{-}}f(x)\) and the right - hand limit \(\lim_{x
ightarrow a^{+}}f(x)\) exist and are equal.
Given \(\lim_{x
ightarrow a^{-}}f(x)=L\) and \(\lim_{x
ightarrow a^{+}}f(x)=M\).
Since \(\lim_{x
ightarrow a}f(x)\) exists, by the definition of the existence of a limit, we must have \(L = M\).

Answer:

A. \(L = M\)