Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).
lim x → -7 f(x)

검증된 단계별 안내

Step 1: Identify the points where the denominator is zero, as these are potential vertical asymptotes. Set the denominator equal to zero: \(x^4 - 49x^2 = 0\).

Step 2: Factor the equation \(x^4 - 49x^2 = 0\) by factoring out \(x^2\), resulting in \(x^2(x^2 - 49) = 0\).

Step 3: Further factor \(x^2 - 49\) using the difference of squares: \((x - 7)(x + 7)\). This gives the factored form \(x^2(x - 7)(x + 7) = 0\).

Step 4: Solve for \(x\) in the equation \(x^2(x - 7)(x + 7) = 0\). The solutions are \(x = 0\), \(x = 7\), and \(x = -7\). These are the potential vertical asymptotes.

Step 5: Analyze the limit \(\lim_{x \to -7} \frac{x + 7}{x^4 - 49x^2}\). Substitute \(x = -7\) into the factored form to check if the limit results in an indeterminate form like \(\frac{0}{0}\), which would confirm a vertical asymptote at \(x = -7\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, evaluating the limit as x approaches -7 helps determine the behavior of the function f(x) near that point, which is crucial for identifying vertical asymptotes.

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity or negative infinity as the input approaches a specific value. They are typically found by identifying values of x that make the denominator zero while the numerator remains non-zero, indicating that the function is undefined at those points.

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions and identify roots or critical points. In the given function, factoring the denominator x^4 - 49x^2 can reveal the values of x that lead to vertical asymptotes, as it allows for easier identification of where the function is undefined.

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).lim x → -7 f(x)