Question

which of the following statements about the function f, if true, cannot be used to conclude that f is defined at x = 1?
a $lim_{x
ightarrow1}f(x)$ exists.
b f is continuous at x = 1.
c f is differentiable at x = 1.
d the line tangent to the graph of f at x = 1 exists.

Explanation:

Step1: Recall function - related concepts

A function \(y = f(x)\) is continuous at \(x=a\) if \(\lim_{x
ightarrow a}f(x)=f(a)\), and differentiable at \(x = a\) implies continuous at \(x=a\), and the existence of the tangent line at \(x=a\) implies differentiable at \(x=a\) (for non - vertical tangent lines).

Step2: Analyze option A

The existence of \(\lim_{x
ightarrow1}f(x)\) does not guarantee that \(f(1)\) is defined. For example, consider the function \(f(x)=\frac{x^{2}-1}{x - 1}\), \(\lim_{x
ightarrow1}f(x)=\lim_{x
ightarrow1}\frac{(x + 1)(x - 1)}{x - 1}=\lim_{x
ightarrow1}(x + 1)=2\), but \(f(1)\) is not defined originally.

Step3: Analyze option B

If \(f\) is continuous at \(x = 1\), then by the definition of continuity \(\lim_{x
ightarrow1}f(x)=f(1)\), so \(f(1)\) is defined.

Step4: Analyze option C

If \(f\) is differentiable at \(x = 1\), then \(f\) is continuous at \(x = 1\) (since differentiability implies continuity). And if \(f\) is continuous at \(x = 1\), \(f(1)\) is defined.

Step5: Analyze option D

The existence of the tangent line to the graph of \(f\) at \(x = 1\) implies that \(f\) is differentiable at \(x = 1\) (for non - vertical tangent lines). Since differentiability implies continuity and continuity implies \(f(1)\) is defined.

Answer:

A. \(\lim_{x
ightarrow1}f(x)\) exists.