Question

The vertical asymptote for \(f(x) = \frac{3x+5{7-x}\) is given by :}

• A. \(x = 7\)
• B. \(x = 3\)
• C. \(x = 5\)
• D. \(x = -1\)

Hint:

To find the vertical asymptote of any rational function:
Simply find the roots of the denominator where the numerator is not zero.
Here, \(7 - x = 0 \implies x = 7\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A vertical asymptote of a rational function \(f(x) = \frac{P(x)}{Q(x)}\) occurs at values of \(x\) where the denominator \(Q(x)\) is zero, provided the numerator \(P(x)\) is non-zero at those same values.
At these points, the function values approach positive or negative infinity.

Step 2: Detailed Explanation:
Let us analyze the given rational function:
\[ f(x) = \frac{3x+5}{7-x} \]
To find the vertical asymptotes, we set the denominator equal to zero:
\[ 7 - x = 0 \implies x = 7 \]
Now, let us verify whether the numerator is non-zero at \(x = 7\):
Substitute \(x = 7\) into the numerator \(P(x) = 3x + 5\):
\[ P(7) = 3(7) + 5 = 21 + 5 = 26 \ne 0 \]
Since the denominator is zero and the numerator is non-zero at \(x = 7\), we can evaluate the one-sided limits to confirm the asymptote:
- As \(x \to 7^-\) (from the left, \(x < 7\)), the denominator \(7-x\) is positive and very small, so \(f(x) \to \infty\).
- As \(x \to 7^+\) (from the right, \(x > 7\)), the denominator \(7-x\) is negative and very small, so \(f(x) \to -\infty\).
Since the function value grows without bound, \(x = 7\) is indeed a vertical asymptote.
This matches Option (A).

Step 3: Final Answer:
The correct option is (A).