Determine each limit (answer as appropriate, with a number, ∞,-∞or does not exist). (a) limx →2^

step 1 Okay, we have the function. We want to find the limit as extra purchase 2. We're going to start

Determine each limit (answer as appropriate, with a number,...

Determine each limit (answer as appropriate, with a number, $\infty,-\infty$ or does not exist).
(a) $\lim _{x \rightarrow 2^{-}} \frac{x-4}{x^{2}-4 x+4}$
(b) $\lim _{x \rightarrow 2^{+}} \frac{x-4}{x^{2}-4 x+4}$
(c) $\lim _{x \rightarrow 2} \frac{x-4}{x^{2}-4 x+4}$

Key Concepts

One-Sided Limits

One-sided limits examine the behavior of a function as its input approaches a particular value from one specific side (either from the left or from the right). This concept is critical when the function behaves differently as it approaches the limit point from different directions, often because of a discontinuity or an asymptote.

Vertical Asymptotes

A vertical asymptote occurs when the value of a rational function increases or decreases without bound as the input approaches a specific point. This is typically due to a division by zero in the function, where the denominator approaches zero and the numerator does not, causing the function to ‘blow up’ towards positive or negative infinity.

Limits of Rational Functions

Evaluating the limits of rational functions involves analyzing the behavior of a fraction where both the numerator and denominator are polynomials. This often requires factoring, simplification, and an investigation into points where the denominator becomes zero, as these points may lead to discontinuities or asymptotic behavior.

Extended Real Number System

In the context of limits, especially when functions grow without bound, the extended real number system is used to express limits that are infinite. Rather than providing a finite real number, limits are sometimes described as positive infinity or negative infinity to indicate unbounded growth in the positive or negative direction.

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