Questions: Find the equations of any vertical asymptotes. f(x) = (x^2+1)/((x^2-1)(x^2-49))

Solution Steps

To find the vertical asymptotes of the function \( f(x) = \frac{x^2 + 1}{(x^2 - 1)(x^2 - 49)} \), we need to determine the values of \( x \) that make the denominator zero, as these are the points where the function is undefined and may have vertical asymptotes. We will set each factor of the denominator equal to zero and solve for \( x \).

The function given is

\[ f(x) = \frac{x^2 + 1}{(x^2 - 1)(x^2 - 49)} \]

The denominator is \((x^2 - 1)(x^2 - 49)\).

To find the vertical asymptotes, we set the denominator equal to zero:

\[ (x^2 - 1)(x^2 - 49) = 0 \]

Solve each factor of the equation separately:

1. \(x^2 - 1 = 0\)

   \[ x^2 = 1 \implies x = \pm 1 \]

2. \(x^2 - 49 = 0\)

   \[ x^2 = 49 \implies x = \pm 7 \]

Final Answer