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 \( x^2 - 25 = 0 \) and solve for x.
2. Step 2: Factor the denominator \( x^2 - 25 \) as a difference of squares: \( (x - 5)(x + 5) = 0 \). This gives the points x = 5 and x = -5 where the function is undefined.
3. Step 3: Determine if these points are vertical asymptotes by checking the behavior of the function as x approaches these values. For x = -5, consider the limit \( \lim_{x \to -5^-} \frac{x - 5}{(x - 5)(x + 5)} \).
4. Step 4: Simplify the expression by canceling the common factor (x - 5) in the numerator and denominator, resulting in \( \lim_{x \to -5^-} \frac{1}{x + 5} \).
5. Step 5: Evaluate the limit \( \lim_{x \to -5^-} \frac{1}{x + 5} \). As x approaches -5 from the left, the denominator approaches 0 from the negative side, indicating a vertical asymptote at x = -5.

Key Concepts

Limits

Limits describe the behavior of a function as the input approaches a certain value. In this context, we are interested in the limit of f(x) as x approaches -5 from the left, which helps determine the function's behavior near that point. Understanding limits is crucial for analyzing continuity and identifying asymptotic behavior.

Vertical Asymptotes

Vertical asymptotes occur at values of x where a function approaches infinity or negative infinity, typically where the denominator of a rational function equals zero while the numerator does not. For the function f(x) = (x − 5) / (x² − 25), we need to find the values of x that make the denominator zero to identify potential vertical asymptotes.

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its factors, which can simplify expressions and help identify roots. In this case, the denominator x² - 25 can be factored into (x - 5)(x + 5), allowing us to easily find the points where the function is undefined and analyze the limits around those points.