Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).
lim x→0 f(x)

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Identify the vertical asymptotes by finding the values of x that make the denominator zero.

Factor the denominator: x^4 - 49x^2 can be rewritten as x^2(x^2 - 49).

Further factor x^2 - 49 as (x - 7)(x + 7) using the difference of squares.

The denominator is now x^2(x - 7)(x + 7). Set each factor equal to zero to find the vertical asymptotes: x^2 = 0, x - 7 = 0, x + 7 = 0.

The vertical asymptotes are at x = 0, x = 7, and x = -7.

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주요 개념

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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 0 helps determine the behavior of the function f(x) near that point, which is crucial for understanding continuity and potential asymptotic behavior.

Vertical Asymptotes

Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value. For the function f(x) = (x + 7) / (x^4 - 49x^2), vertical asymptotes can be found by identifying values of x that make the denominator zero, leading to undefined behavior in the function.

Factoring Polynomials

Factoring polynomials is a technique used to simplify expressions and find roots. In the case of the denominator x^4 - 49x^2, factoring can reveal the critical points where the function may have vertical asymptotes. Recognizing that this expression can be factored as x^2(x^2 - 49) aids in identifying the values of x that lead to undefined behavior.

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x4 − 49x2).lim x→0 f(x)