Location of vertical asymptotes Analyze the following limits...

Location of vertical asymptotes Analyze the following limits and find the vertical asymptotes of $f(x)=\frac{x+7}{x^{4}-49 x^{2}}$
a $.x->7-$
b. $\lim _{x \rightarrow 7^{+}} f(x)$ c. $\lim _{x \rightarrow-7} f(x) \quad$ d. $\lim _{x \rightarrow 0} f(x)$

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Key Concepts

One-Sided Limits

One-sided limits refer to the behavior of a function as the variable approaches a particular value from the left or right. They are crucial for understanding the behavior near vertical asymptotes since the function may approach infinity or negative infinity differently from each direction.

Factoring Polynomials

Factoring is a key algebraic technique used to simplify rational functions by breaking down polynomials into products of simpler factors. This process helps identify zeros of the denominator and numerator, making it easier to analyze asymptotic behavior and discontinuities.

Removable Discontinuities

A removable discontinuity happens when a common factor exists in both the numerator and denominator of a rational function, causing the expression to be simplified. Although the function can be redefined at that point, the original form exhibits a hole rather than a vertical asymptote.

Vertical Asymptotes

Vertical asymptotes occur in rational functions when the denominator is zero and the numerator is nonzero, leading to unbounded behavior as x approaches that value. They are identified by finding the zeros of the denominator that do not cancel with zeros in the numerator.

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