if \\( \\lim _{x \\rightarrow a^{-}} f(x)=l \\) and \\( \\lim _{x \\rightarrow a^{+}} f(x)=m \\), where \\(…

if \\( \\lim _{x \\rightarrow a^{-}} f(x)=l \\) and \\( \\lim _{x \\rightarrow a^{+}} f(x)=m \\), where \\( l \\) and \\( m \\) are finite real numbers, then how are \\( l \\) and \\( m \\) related if \\( \\lim _{x \\rightarrow a} f(x) \\) exists?\n\nchoose the correct answer.\n\n\\( \\bigcirc \\) a. \\( l = m \\)\n\\( \\bigcirc \\) b. \\( l \\neq m \\)\n\\( \\bigcirc \\) c. \\( l < m \\)\n\\( \\bigcirc \\) d. \\( l > m \\)

if \\( \\lim _{x \\rightarrow a^{-}} f(x)=l \\) and \\( \\lim _{x \\rightarrow a^{+}} f(x)=m \\), where \\( l \\) and \\( m \\) are finite real numbers, then how are \\( l \\) and \\( m \\) related if \\( \\lim _{x \\rightarrow a} f(x) \\) exists?\n\nchoose the correct answer.\n\n\\( \\bigcirc \\) a. \\( l = m \\)\n\\( \\bigcirc \\) b. \\( l \\neq m \\)\n\\( \\bigcirc \\) c. \\( l < m \\)\n\\( \\bigcirc \\) d. \\( l > m \\)

Answer

Answer:

A. (L = M)

Explanation:

Step1: Recall the limit existence condition

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