91Ó°ÊÓ

In calculus, sometimes we are interested in understanding how a function behaves as it approaches a specific point from one side only. This is known as a one-sided limit. We have the limit from the right, often written as \( \lim_{x \to a^+} f(x) \), and the limit from the left, \( \lim_{x \to a^-} f(x) \). These symbols indicate we are exploring the behavior of the function as \( x \) gets closer to \( a \) from values larger than \( a \), or smaller than \( a \), respectively.
In the given exercise, we found two one-sided limits for the function \( f(x) = \frac{2}{x^{1/5}} \) as \( x \) approaches 0 both from positive and negative directions.

For the limit from the positive side, as \( x \to 0^+ \), the denominator becomes very small, leading to an infinitely large positive value, hence the limit is \( +\infty \).
Conversely, when \( x \to 0^- \), the denominator being a small negative number causes the fraction to tend to a large negative value, thus \( -\infty \).

• Right-hand limits look at approaching values from larger numbers.
• Left-hand limits examine from smaller numbers.
• These help understand sharp changes or undefined points.

Infinite limits describe situations where a function's value increases or decreases without bound as it approaches a specific point. An infinite limit doesn't give a finite number but rather shows the function's behavior as tending to infinity.
In the problem, we explored the limits of \( f(x) = \frac{2}{x^{1/5}} \) as \( x \) approaches 0. Here, because the denominator approaches 0, the overall fraction grows or shrinks without limit depending on the direction from which \( x \) approaches zero.

These infinite limits can help us understand vertical asymptotes in a function's graph. A vertical asymptote occurs where the function reaches exceedingly large values, much like \( x = 0 \) for the given function.

• Infinite limits approach large magnitudes.
• They indicate vertical asymptotes or sharp changes.
• Essential in analyzing function behavior around critical points.

Examining how functions behave as they near a particular point is crucial in calculus. This analysis can uncover much about the nature of the function at that point, including rates of change and potential discontinuities.
The exercise focused on understanding the behavior as \( x \) nears zero for the function \( f(x) = \frac{2}{x^{1/5}} \). The fifth root function behaves uniquely:

• As \( x \to 0^+ \), \( x^{1/5} \to 0^+ \), leading \( f(x) \) to grow towards \( +\infty \). This shows a sharp vertical asymptote.
• As \( x \to 0^- \), \( x^{1/5} \) also approaches 0 but is negative, thus \( f(x) \to -\infty \), indicating an opposite vertical asymptote.
• This differing behavior from each side reflects a point of discontinuity.

Understanding how a function changes around critical points can help in designing graphs and predicting potential real-world applications where sudden changes occur.