Question

Suppose the function f is continuous at the point P(c, f(c)). Then, the graph of f has a vertical tangent at P if :

• A. $\lim_{x\to \text{c}^-} f'(x) = +\infty$ and $\lim_{x\to \text{c}^+} f'(x) = +\infty$
• B. $\lim_{x\to \text{c}^-} f'(x) = +\infty$ and $\lim_{x\to \text{c}^+} f'(x) = -\infty$
• C. $\lim_{x\to \text{c}^-} f'(x) = -\infty$ and $\lim_{x\to \text{c}^+} f'(x) = +\infty$
• D. $\lim_{x\to \text{c}^-} f'(x) = 0$ and $\lim_{x\to \text{c}^+} f'(x) = +\infty$

Hint:

For a vertical tangent, both the left-hand and right-hand limits of $f'(x)$ must approach the same infinity ($\pm \infty$). If they approach opposite infinities (one goes to $+\infty$ and the other to $-\infty$), the graph has a vertical cusp, not a vertical tangent.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A curve $y = f(x)$ has a vertical tangent at a point $P(c, f(c))$ if the function is continuous at $c$ and the slope of the tangent line approaches infinity from both sides of $c$.
Detailed Explanation:
Let us review the mathematical conditions for a vertical tangent:
- The slope of the tangent line is represented by the derivative $f'(x)$.
- For the tangent line to be vertical, its slope must be undefined in a specific way: the slope must become infinitely steep, meaning $|f'(x)| \to \infty$ as $x \to c$.
- Specifically, the left-hand limit and the right-hand limit of the derivative must both approach either positive infinity ($+\infty$) or both approach negative infinity ($-\infty$). This ensures the tangent line has a consistent vertical direction at the point.
- Comparing this with the options, Option (A) states: \[ \lim_{x\to \text{c}^-} f'(x) = +\infty \quad \text{and} \quad \lim_{x\to \text{c}^+} f'(x) = +\infty \] This meets the mathematical requirement for a vertical tangent.

Step 2: Final Answer:
The condition matches Option (A).