Analyze the following limits and find the vertical...

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

Key Concepts

Limits

Limits are fundamental to calculus and analysis, describing the behavior of functions as the input approaches a particular value. They are especially useful for understanding the behavior of functions near points where they are not easily computed directly, such as points of discontinuity or where the function tends to infinity.

Vertical Asymptotes

Vertical asymptotes occur where a function’s value increases or decreases without bound as the input approaches a particular value. They indicate locations in the domain where the function is undefined, typically due to division by zero, which cannot be remedied by simple cancellation.

Removable versus Non-Removable Discontinuities

Discontinuities in a function can be classified as removable or non-removable. A removable discontinuity, often referred to as a 'hole', occurs when a common factor cancels out in the numerator and denominator, while non-removable discontinuities, such as vertical asymptotes, remain after cancellation and cause the function to diverge at specific points.

Polynomial Factoring

Polynomial factoring involves expressing a polynomial as a product of its simpler factors. This technique is critical when evaluating limits and identifying discontinuities, as it can reveal shared factors between the numerator and denominator (leading to removable discontinuities) and isolated factors in the denominator that indicate potential vertical asymptotes.

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