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When we talk about one-sided limits in calculus, we're referring to the limit of a function as the input approaches a particular value from just one side—either from the left or the right. To understand one-sided limits, consider approaching a specific point on the x-axis and checking how the function behaves just before and just after reaching that point.

In the given exercise, the function is \[ \lim _{x \rightarrow 1} \frac{x}{x-1} \]. We examine the behavior as x approaches 1 from the left side (noted as \(x \to 1^-\), meaning less than 1) and from the right side (noted as \(x \to 1^+\), meaning greater than 1).

• For \(x \to 1^-\), when x is slightly less than 1, the denominator (x-1) is a small negative number. Consequently, the fraction becomes very negative and the one-sided limit is \(-\infty\).
• For \(x \to 1^+\), when x is slightly more than 1, the denominator (x-1) is a small positive number, making the fraction very positive. Thus, the one-sided limit is \(+\infty\).

Understanding one-sided limits helps in analyzing the exact behavior of a function at a specific point, especially near discontinuities.

Infinite limits occur when the value of a function becomes infinitely large as the input approaches a specific point. These can either be positively infinite, meaning the function grows without bounds, or negatively infinite, where the function decreases without bounds.

In our exercise, as x gets closer to 1, the denominator (x-1) tends towards zero. However, since the numerator x remains close to a fixed value (1), the result is that our function \( \frac{x}{x-1} \) tends towards either positive or negative infinity depending on the direction of approach.

• If x approaches 1 from the left (\(x \to 1^-\)), the function trends towards \(-\infty\)
• If x approaches 1 from the right (\(x \to 1^+\)), the function trends towards \(+\infty\)

The notion of infinite limits is fundamental in assessing how functions behave as they approach critical points, which is crucial in determining the asymptotic behavior of functions.

Discontinuities in a function tell us that the function makes a leap or has a break in its graph at a certain point. Understanding the behavior of functions near these discontinuities helps in predicting how a function might behave overall. At a discontinuity, limits might not exist, but one-sided and infinite limits can still give insights.

For the function \( \frac{x}{x-1} \), there's a discontinuity at \(x=1\). As x approaches 1, the denominator approaches zero, causing the function value to spike aggressively either towards positive or negative infinity, depending on the direction.

• As \(x \to 1^-\), the function races to \(-\infty\)
• As \(x \to 1^+\), it soars to \(+\infty\)

These insights help us understand that despite the graph being undefined at \(x=1\), evaluating the nearby behavior of the function is pivotal for grasping its overall characteristics and anticipating its behavior in different contexts.