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Chapter 12: Problem 20

Using a Graphing Utility to Estimate a Limit In Exercises \(11-22,\) create a table of values for the function and use the result to estimate the limit numerically. Use a graphing utility to graph the corresponding function to confirm your result graphically. $$\lim _{x \rightarrow 0} \frac{1-e^{-4 x}}{x}$$

Short Answer

Expert verified
The limit as x approaches 0 of \( \frac{1 - e^{-4x}}{x} \) is 4.

Step by step solution

01

Apply L'Hopital's Rule

L'Hopital's Rule states that the limit of a quotient of two functions as x approaches a certain value, where both functions approach 0 or both approach infinity, is equal to the limit of the quotients of their derivatives. So, we compute the derivatives of the numerator and the denominator. The derivative of \(1-e^{-4x}\) is \(4e^{-4x}\) and the derivative of \(x\) is \(1\).
02

Compute the New Limit

After applying L'Hopital's Rule, the limit becomes: \(\lim_{x \rightarrow 0} \frac{4e^{-4x}}{1}\). This simplifies to: \(4 \lim_{x \rightarrow 0} e^{-4x}\). As \(x\) approaches 0, \(e^{-4x}\) approaches 1, since any exponential function with power 0 is 1. So, our limit becomes \(4 * 1 = 4\).
03

Confirm Result Graphically

Using a graphing utility, graph the function \(y = \frac{1 - e^{-4x}}{x}\) around \(x = 0\). The graph should approach a value of 4, confirming our previously calculated limit.

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Key Concepts

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

Numerical Estimation
Sometimes, understanding the behavior of a function as it approaches a certain value can be tricky. This is where numerical estimation becomes useful. The concept involves creating a table of values for the function around the point of interest, in this case, as \(x\) approaches 0.
By choosing very small values near zero, both positive and negative, you can observe the corresponding outputs of the function \(\frac{1-e^{-4x}}{x}\). As these outputs get closer to each other, they provide an estimate of the limit.
This method is a hands-on approach allowing you to see the gradual change in values as \(x\) gets closer to 0. It serves as an intuitive check before delving into more complex analytical methods.
Graphing Utilities
Graphing utilities are powerful tools for visualizing mathematical functions and their behavior at specific points. When estimating limits graphically, such as with \(\lim_{x \rightarrow 0} \frac{1-e^{-4x}}{x}\), a graphing calculator or software can be employed.
By entering the function into the graphing utility, you can plot the curve near \(x = 0\). Observe how the graph behaves. If it approaches a horizontal line, this line's value can give insight into the limit.
This visual confirmation complements numerical estimations, adding another layer of understanding, and is particularly useful when analytical solutions are challenging to grasp immediately.
Exponential Functions
Exponential functions, like \(e^{-4x}\), play a significant role in calculus and limit problems. An exponential function generally has the form \(e^{kx}\) where \(e\) is Euler鈥檚 number, approximately 2.718. It's a constantly growing function unless the exponent is negative, in which case it decays.
In the problem \(\lim_{x \rightarrow 0} \frac{1-e^{-4x}}{x}\), \(e^{-4x}\) approaches 1 as \(x\) moves towards 0. This is because when any number to the power of 0 is 1.
Understanding how exponential functions behave around different points helps you predict how the function will act and aids in calculating limits like these by making the process straightforward.

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Most popular questions from this chapter

Determine whether the statement is true or false. Justify your answer. The slope of the graph of \(y=x^{2}\) is different at every point on the graph of \(f .\)

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