Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^- f(x) and lim x→a⁺ f(x).
f(x) = (x^4−1)/(x^2−1)

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To find the vertical asymptotes, set the denominator equal to zero: \(x^2 - 1 = 0\). Solve for \(x\) to find the points where the function is undefined.

This can be factored as \((x - 1)(x + 1) = 0\). Therefore, the solutions are \(x = 1\) and \(x = -1\). These are the potential vertical asymptotes.

Check if the numerator \(x^4 - 1\) can be factored to cancel any common factors with the denominator. Factor \(x^4 - 1\) as \((x^2 - 1)(x^2 + 1)\).

The expression becomes \(\frac{(x^2 - 1)(x^2 + 1)}{x^2 - 1}\). Cancel the common factor \(x^2 - 1\), leaving \(f(x) = x^2 + 1\) for \(x \neq \pm 1\).

For \(x = 1\) and \(x = -1\), analyze \(\lim_{x \to 1^-} f(x)\), \(\lim_{x \to 1^+} f(x)\), \(\lim_{x \to -1^-} f(x)\), and \(\lim_{x \to -1^+} f(x)\) using the simplified function \(f(x) = x^2 + 1\).

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Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value from either the left or the right. This typically happens at points where the function is undefined, often due to division by zero. Identifying vertical asymptotes involves finding the values of x that make the denominator zero while ensuring the numerator is not also zero at those points.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, we analyze the left-hand limit (as x approaches a from the left) and the right-hand limit (as x approaches a from the right) to determine the behavior of the function near the asymptote. This helps in understanding whether the function tends to positive or negative infinity.

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions, making it easier to analyze their behavior, such as finding asymptotes. In the given function f(x) = (x^4−1)/(x^2−1), factoring both the numerator and denominator can reveal common factors and help identify points of discontinuity. This process is crucial for determining where vertical asymptotes may exist and for simplifying the function before evaluating limits.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).f(x) = (x^4−1)/(x^2−1)