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Chapter 2: Problem 46

Find the limits in Exercises \(37-48\) $$ { a. }\lim _{x \rightarrow 0^{+}} \frac{2}{x^{1 / 5}} \quad \text { b. } \lim _{x \rightarrow 0} \frac{2}{x^{1 / 5}} $$

Short Answer

Expert verified
a. Infinity, b. Does not exist.

Step by step solution

01

Understand the Expression

The expression given is \( \lim_{x \to 0^{+}} \frac{2}{x^{1/5}} \) for part a and \( \lim_{x \to 0} \frac{2}{x^{1/5}} \) for part b.
02

Analyzing Part a

For \( \lim_{x \to 0^{+}} \frac{2}{x^{1/5}} \), as \( x \to 0^{+} \), \( x^{1/5} \) approaches 0 from the positive side. So, \( \frac{1}{x^{1/5}} \) becomes very large, and consequently, \( \frac{2}{x^{1/5}} \) approaches infinity.
03

Analyzing Part b

For \( \lim_{x \to 0} \frac{2}{x^{1/5}} \), we consider the approach from both sides. As \( x \to 0^{+} \), the limit is similar to part a and approaches infinity. As \( x \to 0^{-} \), \( x^{1/5} \) would be a small negative number, causing \( \frac{2}{x^{1/5}} \) to approach negative infinity. Thus, the two one-sided limits do not match and the limit does not exist for \( x \to 0 \) without a specified sign.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-Sided Limits
One-sided limits focus on the behavior of a function as the input values approach a particular point from one side only. In the context of our exercise, we were interested in the limit of \( \lim_{x \to 0^{+}} \frac{2}{x^{1/5}} \). This means we need to observe what happens to the function \( \frac{2}{x^{1/5}} \) when \( x \) approaches zero from the positive side.
  • If the limit approaches from the positive side, consider all values slightly larger than the point.
  • If it approaches from the negative side, consider values just smaller than the point.
For \( \lim_{x \to 0^{+}} \), as \( x \) gets closer to zero from the right, the term \( x^{1/5} \) becomes very small but remains positive. This causes the expression \( \frac{2}{x^{1/5}} \) to grow very large, leading us to infinity.
Infinite Limits
Infinite limits describe situations where a function grows larger and larger without bound as the input approaches a certain value. In our exercise, \( \lim_{x \to 0^{+}} \frac{2}{x^{1/5}} \) demonstrates an infinite limit.
  • When a limit is infinite, the function behavior is described as going towards positive or negative infinity.
  • Infinite limits often occur with functions that involve division by expressions approaching zero.
As this limit approaches zero from the positive side and \( x^{1/5} \) gets tiny, \( \frac{2}{x^{1/5}} \) shoots off towards infinity. It is always useful to graph these functions to see their growth visually, which provides a clearer understanding of how they behave near the given point.
Power Functions
Power functions like \( x^{1/5} \) involve variables raised to specific powers or fractions of powers. These functions exhibit unique behaviors especially near zero or infinity.
  • Fractional powers, such as \( x^{1/5} \), can represent roots.
  • Near zero, such power functions tend to change rapidly, affecting overall function behavior.
As seen in our exercise, at \( x \to 0 \), both positive and negative sides need to be considered for power functions. Each side behaves differently, as fractional powers can take on both real positive and negative values based on the sign of \( x \). Drawing small segments on a graph can showcase these rapid changes, helping to visualize the outcome of limits involving power functions.
Nonexistent Limits
A limit does not exist when the values from both sides of an approach point lead to distinctly different results. This is what happened in part b of our exercise, \( \lim_{x \to 0} \frac{2}{x^{1/5}} \).
  • For limits to exist, one-sided limits need to converge to the same value from both directions.
  • Inconsistent results on both sides signify a non-existent limit.
In our case, approaching zero from the positive side leads to infinity, while from the negative side leads to negative infinity. This mismatch in results confirms that the limit does not exist. When students encounter non-existent limits, it's essential to check both directions and validate that outcomes differ to declare the limit nonexistent.

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