Question

If ${\lim }_{x\to a+}f(x)=l={\lim }_{x\to a-}g(x)$ and ${\lim }_{x\to a-}f(x)=m={\lim }_{x\to a+}g(x)$ , then the function $f(x).g(x)$

• A. is continuous at $x=a$
• B. is not continuous at $x=a$.
• C. has a limit when $x\to a$ and it is equal to $Im$.
• D. has a limit when $x\to a$ a but it is not equal to $Im$.

Answer 1

Answer: (C)

${\lim }_{x\to a-}\left[f\left(x\right)-g\left(x\right)\right]={\lim }_{x\to a-}f\left(x\right)={\lim }_{x\to a-}g\left(x\right)=ml$
${\lim }_{x\to a+}\left[f\left(x\right)-g\left(x\right)\right]={\lim }_{x\to a+}f\left(x\right)-{\lim }_{x\to a+}g\left(x\right)=lm$
$\therefore {\lim }_{x\to a}f\left(x\right)g\left(x\right)=lm$