Analyze the following limits and find the vertical asymptotes of f(x) =(x − 5) / (x² − 25).
lim x→−5⁺ f(x)

Verified step by step guidance

1. Step 1: Identify the points where the function f(x) = \( \frac{x - 5}{x^2 - 25} \) is undefined. This occurs when the denominator is zero. Set the denominator equal to zero: \( x^2 - 25 = 0 \).
2. Step 2: Solve the equation \( x^2 - 25 = 0 \) to find the values of x that make the denominator zero. This can be factored as \( (x - 5)(x + 5) = 0 \), giving the solutions \( x = 5 \) and \( x = -5 \).
3. Step 3: Determine if these points are vertical asymptotes by checking the behavior of the function as x approaches these values. Since the numerator \( x - 5 \) is zero at \( x = 5 \), the function has a removable discontinuity at \( x = 5 \), not a vertical asymptote.
4. Step 4: Analyze the limit \( \lim_{x \to -5^+} f(x) \). As x approaches -5 from the right, the denominator \( x^2 - 25 \) approaches zero, and the numerator \( x - 5 \) approaches -10. This indicates a vertical asymptote at \( x = -5 \).
5. Step 5: Conclude that the function has a vertical asymptote at \( x = -5 \) because the limit of f(x) as x approaches -5 from the right is either positive or negative infinity.