Question

if $lim_{x
ightarrow a}f(x)=f(a)$, then which of the following statements about $f$ must be true?
a. $f$ is continuous at $x = a$.
b. $f$ is differentiable at $x = a$.
c. for all values of $x$, $f(x)=f(a)$.
d. the line $y = f(a)$ is tangent to the graph of $f$ at $x = a$.
e. the line $x = a$ is a vertical asymptote of the graph of $f$.

Explanation:

The definition of continuity of a function \(y = f(x)\) at \(x=a\) is \(\lim_{x
ightarrow a}f(x)=f(a)\). If the limit of the function as \(x\) approaches \(a\) is equal to the value of the function at \(x = a\), the function is continuous at \(x=a\). Differentiability requires the existence of the limit \(\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}\), which is not guaranteed just by \(\lim_{x
ightarrow a}f(x)=f(a)\). Option C is incorrect as it implies the function is a constant function. For the line \(y = f(a)\) to be tangent to the graph of \(f\) at \(x = a\), the derivative \(f^{\prime}(a)=0\) which is not implied. A vertical - asymptote at \(x = a\) means \(\lim_{x
ightarrow a}f(x)=\pm\infty\) which is not the case here.

Answer:

A. \(f\) is continuous at \(x = a\)