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Chapter 3: Problem 50

Determine graphically and numerically whether or not any of the following limits exist. a. \(\lim _{x \rightarrow 0^{-}}\left(x \cdot 3^{1 / x}\right)\) b. \(\lim _{x \rightarrow 0^{+}}\left(x \cdot 3^{1 / x}\right)\) c. \(\lim _{x \rightarrow 0}\left(x \cdot 3^{1 / x}\right)\)

Short Answer

Expert verified
a. \( -\infty \); b. \( +\infty \); c. Does not exist.

Step by step solution

01

Understand the Function Behavior

The function in question is \( f(x) = x \cdot 3^{1/x} \). We need to determine its behavior as \( x \) approaches 0 from the left (\( 0^- \)), from the right (\( 0^+ \)), and finally from both sides (\( 0 \)).
02

Evaluate the Limit as x Approaches 0 from the Left

Consider \( \lim_{x \to 0^-} (x \cdot 3^{1/x}) \). For values of \( x \) approaching 0 from the negative side, \( 3^{1/x} \) grows very large, since \( 1/x \) becomes very negative, making the exponent positive and large. The product \( x \cdot 3^{1/x} \) includes a negative \( x \), resulting in the limit tending to, negatively, infinity. Thus, \( \lim_{x \to 0^-} (x \cdot 3^{1/x}) = -\infty \).
03

Evaluate the Limit as x Approaches 0 from the Right

Consider \( \lim_{x \to 0^+} (x \cdot 3^{1/x}) \). For values of \( x \) approaching 0 from the positive side, \( 3^{1/x} \) again is very large, given \( 1/x \) is very large. Here, \( x \) is positive, so the product \( x \cdot 3^{1/x} \) becomes positively infinite. Hence, \( \lim_{x \to 0^+} (x \cdot 3^{1/x}) = +\infty \).
04

Evaluate the Overall Limit as x Approaches 0

Since the limits from the left \( (-\infty) \) and from the right \( (+\infty) \) are different, \( \lim_{x \to 0} (x \cdot 3^{1/x}) \) does not exist. For a two-sided limit to exist, both one-sided limits must be equal, but in this case, they are not. Therefore, the limit does not exist as \( x \) approaches 0.

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Key Concepts

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

Graphical Analysis
When trying to understand limits graphically, visualizing the behavior of the function on a graph is essential. By plotting the function \( f(x) = x \cdot 3^{1/x} \), you can observe how the function behaves as \( x \) approaches zero from both the left and right sides. In the graphical representation:
  • As \( x \) approaches \( 0^- \) (from the left), the graph shows the function decreasing sharply, heading towards negative infinity.
  • As \( x \) approaches \( 0^+ \) (from the right), the graph ascends steeply towards positive infinity.

By analyzing the graph, it's clear that the two paths (left and right approaching zero) do not converge to the same point. This indicates that the limit as \( x \) approaches zero does not exist since there is a significant discrepancy in direction and behavior from either side.
Numerical Analysis
Evaluating a limit numerically involves calculating the function's values close to the point of interest. Let's look specifically at \( f(x) = x \cdot 3^{1/x} \) and examine points near zero.
  • Approaching from \( x = -0.1, -0.01, -0.001, \ldots \) results in increasingly negative large values, indicating a trend towards negative infinity.
  • Approaching from \( x = 0.1, 0.01, 0.001, \ldots \) gives increasingly large positive numbers, suggesting a trend towards positive infinity.

These numerical insights reinforce that as \( x \) approaches zero, the function values from left and right differ vastly, showing they do not meet at a common value. Thereby proving the overall limit is not defined, as the results from both sides are contradictory.
One-Sided Limits
One-sided limits focus on the behavior of a function from only one direction, either the left or the right. For the function \( f(x) = x \cdot 3^{1/x} \), the limits are:
  • Left Side (\( x \to 0^- \)): As \( x \) becomes negative and approaches zero, the term \( 3^{1/x} \) becomes a very large positive number, turning \( x \cdot 3^{1/x} \) into negative infinity.
  • Right Side (\( x \to 0^+ \)): Near zero on the positive side, \( 3^{1/x} \) is again extremely large, and combined with positive \( x \), the product heads towards positive infinity.

Understanding these one-sided limits is crucial. They tell us that since the function heads towards different infinities from the left and right, the two-sided limit at \( x = 0 \) cannot exist.
Function Behavior
The behavior of a function describes how it acts over its domain or as it nears particular values. For \( f(x) = x \cdot 3^{1/x} \), observe how it behaves as \( x \) nears zero:
  • The product involves \( x \) and an exponential component, \( 3^{1/x} \), which dominates as \( x \) nears zero.
  • This exponential component grows very large as \( 1/x \) skyrockets when \( x \) is almost zero, creating extreme output values based on whether \( x \) is positive or negative.

The behavior at the point of interest (0) shows a mismatch. From the left and right, vastly different outputs (negative and positive infinity, respectively) show the behavior does not resolve at a single limit value at zero. This complexity is what prevents the limit from existing seamlessly.

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