91Ó°ÊÓ

One-sided limits help us find the behavior of a function as the input approaches a certain point from one specific direction - either from the left (denoted as \(x \to c^-\) or from the right (denoted as \(x \to c^+\)). For the given problem, we need to examine the limit as \(x\) approaches 4 from both directions to fully understand the function \(\frac{x}{x-4}\).
When \(x\) approaches 4 from the left side (\(x \to 4^-\)), the denominator \(x-4\) is negative yet small, which means the fraction becomes a large negative number (tending towards negative infinity).
Similarly, when \(x\) approaches 4 from the right side (\(x \to 4^+\)), the denominator \(x-4\) is positive yet small, causing the fraction to become a large positive number (tending towards positive infinity).
This distinction highlights the importance of one-sided limits in determining the differing behaviors of a function near a point from each direction.

Infinite limits occur when a function increases or decreases without bound as the input approaches a particular value. In the given exercise, as \(x\) approaches 4 for the function \(\frac{x}{x-4}\), either from the right or from the left, the value of the function becomes unbounded (i.e., approaches infinity or negative infinity).
From the left side (\(x \to 4^-\)), the function \(\frac{x}{x-4}\) tends towards negative infinity, which can be written as \(\lim _{x \to 4^-} \frac{x}{x-4} = -\infty\).
From the right side (\(x \to 4^+\)), it tends towards positive infinity, expressed as \(\lim _{x \to 4^+} \frac{x}{x-4} = +\infty\).
Understanding infinite limits helps us interpret how a function behaves near certain critical points.

When evaluating limits, the goal is to find the value that a function approaches as the input approaches a certain point. To evaluate the limit \(\lim _{x \rightarrow 4} \frac{x}{x-4}\), we first try direct substitution. Substituting \(x = 4\) results in \(\frac{4}{4-4}\), an undefined form because it gives \(\frac{4}{0}\). This indicates the limit doesn't exist in a finite sense.
Next, observing the behavior of the function near \(x = 4\) from both sides (as detailed in one-sided and infinite limits), reveals that the function approaches different infinities depending on the direction. Thus, the two-sided limit doesn't exist because the function does not approach the same value from the left and right.
In summary, the steps for evaluating limits are:

• Attempt direct substitution.
• Analyze one-sided behavior.
• Understand the function's behavior near the point of interest.