Problem 34
Question
Use a table and a graph to find out what happens to $$ f(x)=\exp \left(\frac{1}{x}\right) $$ as \(x \rightarrow 0\)
Step-by-Step Solution
Verified Answer
As \( x \rightarrow 0^+ \), \( f(x) \rightarrow 1 \); as \( x \rightarrow 0^- \), \( f(x) \rightarrow \\infty \).
1Step 1: Understand the behavior of the function
The function given is \( f(x) = \exp(\frac{1}{x}) \). We need to examine the behavior of \( f(x) \) as \( x \) approaches 0 from both the positive and negative sides.
2Step 2: Create a table of values
Generate a table of \( x \) values approaching 0 from both directions, for example: \( x = -0.1, -0.01, 0, 0.01, 0.1 \). Calculate the corresponding \( f(x) \) values:- \( x = -0.1 \), \( f(x) \approx 2.69 \times 10^{43} \)- \( x = -0.01 \), \( f(x) \approx 7.38 \times 10^{435} \)- \( x = 0 \), \( f(x) \) is undefined.- \( x = 0.01 \), \( f(x) \approx 1.22 \)- \( x = 0.1 \), \( f(x) \approx 1.11 \)
3Step 3: Plot the graph
Create a graph with \( x \) on the horizontal axis and \( f(x) \) on the vertical axis, using the table of values. Note the sharp rise in \( f(x) \) as \( x \) approaches 0 from the negative side, and a gradual approach to 1 from the positive side.
4Step 4: Analyze the results
From the table and graph, observe that as \( x \rightarrow 0^- \), \( f(x) \rightarrow \infty \) (approaches infinity), and as \( x \rightarrow 0^+ \), \( f(x) \rightarrow 1 \). Hence, the function shows drastically different behavior from the left and right of 0.
Key Concepts
Exponential functionLimitsGraphical analysis
Exponential function
An exponential function is a mathematical expression that involves a constant base raised to a variable exponent. In our exercise, the function given is \( f(x) = \exp\left(\frac{1}{x}\right) \). Here, \( \exp\) refers to the exponential function, which is equivalent to \( e^{\text{something}} \), where \( e \) is Euler's number, approximately 2.71828.
The exponential function has unique properties:
Understanding these properties can help when analyzing complex functions like the one in the exercise. Here, the exponent \( \frac{1}{x} \) dramatically changes the behavior of \( f(x) \) as \( x \) approaches zero. For negative \( x \), \( \frac{1}{x} \) becomes a large negative number, leading to the exponential growth of \( f(x) \). Conversely, for positive \( x \), \( \frac{1}{x} \) becomes large and positive, causing \( f(x) \) to decrease toward 1.
This distinction is crucial for solving problems involving exponential behavior.
The exponential function has unique properties:
- It rapidly increases or decreases as the variable in the exponent grows.
- When the exponent is positive, \( e^x \) grows larger as \( x \) increases.
- When the exponent is negative, \( e^x \) becomes a fraction and approaches zero as \( x \) decreases further.
Understanding these properties can help when analyzing complex functions like the one in the exercise. Here, the exponent \( \frac{1}{x} \) dramatically changes the behavior of \( f(x) \) as \( x \) approaches zero. For negative \( x \), \( \frac{1}{x} \) becomes a large negative number, leading to the exponential growth of \( f(x) \). Conversely, for positive \( x \), \( \frac{1}{x} \) becomes large and positive, causing \( f(x) \) to decrease toward 1.
This distinction is crucial for solving problems involving exponential behavior.
Limits
The concept of limits is fundamental in calculus. It helps us understand how a function behaves as the input approaches a certain value. In this exercise, we are interested in the limit of \( f(x) = \exp\left(\frac{1}{x}\right) \) as \( x \to 0 \).
First, let's break it down. Calculating the limit as \( x \to 0^- \) (from the left) means we observe the function as \( x \) approaches zero from the negative side:
Now, as \( x \to 0^+ \) (from the right):
Limits show how different approaches to zero significantly impact the function. This tells us that the function is not defined at \( x = 0 \), as it presents a discontinuity: it doesn't settle on a single value but diverges based on the direction from which zero is approached.
First, let's break it down. Calculating the limit as \( x \to 0^- \) (from the left) means we observe the function as \( x \) approaches zero from the negative side:
- Here, \( \frac{1}{x} \) becomes very negative, thus \( \exp\left(\frac{1}{x}\right) \) grows incredibly fast and tends towards infinity.
Now, as \( x \to 0^+ \) (from the right):
- \( \frac{1}{x} \) becomes a large positive quantity, making \( \exp\left(\frac{1}{x}\right) \) approach 1.
Limits show how different approaches to zero significantly impact the function. This tells us that the function is not defined at \( x = 0 \), as it presents a discontinuity: it doesn't settle on a single value but diverges based on the direction from which zero is approached.
Graphical analysis
Graphical analysis provides a visual understanding of a function's behavior by plotting it on a graph. For the function \( f(x) = \exp\left(\frac{1}{x}\right) \), a graph can visually highlight how \( f(x) \) changes as \( x \) approaches zero.
To construct this graph:
This graphical approach reveals:
To construct this graph:
- Plot points for values like \( x = -0.1, -0.01, 0.01, 0.1 \), using the corresponding function values found earlier.
- For negative values close to zero, you'll observe that \( f(x) \) shoots upward rapidly, indicating a vertical asymptote as \( x \to 0^- \).
- For positive values approaching zero, \( f(x) \) nivelates towards 1.
This graphical approach reveals:
- The drastic difference in behavior of \( f(x) \) as \( x \) approaches zero from different sides.
- The asymptotic nature of the function at \( x = 0 \), where it's undefined, with \( f(x) \to \infty \) as \( x \to 0^- \) and \( f(x) \to 1 \) as \( x \to 0^+ \).
Other exercises in this chapter
Problem 33
Use a table and a graph to find out what happens to $$ f(x)=\frac{2}{x}-\frac{1}{x^{2}} $$ as \(x \rightarrow 0^{-}\). What happens as \(x \rightarrow 0^{+}\) ?
View solution Problem 34
In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-2} \sqrt{6+x} $$
View solution Problem 35
In Problems 29-48, find the limits. $$ \lim _{x \rightarrow-1} \sqrt{x^{2}+2 x+2} $$
View solution Problem 35
Use a graphing calculator to investigate $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$
View solution