Problem 56 - Calculus Early Transcendentals

Question

Determine the interval(s) on which the following functions are continuous; then analyze the given limits. $$f(x)=\frac{e^{2 x}-1}{e^{x}-1} ; \lim _{x \rightarrow 0} f(x)$$

Step-by-Step Solution

Answer

Question: Determine the interval(s) of continuity for the function $$f(x) = \frac{e^{2x} - 1}{e^x - 1}$$ and calculate the limit $$\lim_{x\to 0} f(x)$$. Answer: The function $$f(x) = \frac{e^{2x} - 1}{e^x - 1}$$ is continuous on the intervals $$(-\infty, 0)$$ and $$(0, \infty)$$. The limit $$\lim_{x \rightarrow 0} f(x) = 2$$.

Step 1: Find the discontinuities of the function

To find the values of $$x$$ for which the denominator is zero, we'll set $$e^x - 1$$ equal to zero and solve for $$x$$. $$e^x - 1 = 0$$ Adding 1 to both sides of the equation: $$e^x = 1$$ Taking the natural logarithm of both sides: $$x = ln(1)$$ $$x = 0$$ So the function has a discontinuity at $$x = 0$$.

Step 2: Determine the intervals of continuity

As we've determined that the only discontinuity is at $$x = 0$$, the intervals of continuity are just all real numbers except for this point. So the function is continuous on the intervals $$(-\infty, 0)$$ and $$(0, \infty)$$.

Step 3: Analyze the given limit

To find the limit of the function as $$x$$ approaches 0, we will use L'Hôpital's rule. Since the function is in the indeterminate form $$\frac{0}{0}$$ when $$x=0$$, we can apply L'Hôpital's rule by taking the derivative of the numerator and denominator with respect to $$x$$ and finding the limit of the resulting function. $$\lim_{x \rightarrow 0} f(x) = \lim_{x \rightarrow 0} \frac{e^{2x} - 1}{e^x - 1}$$ Taking the derivative of the numerator: $$\frac{d}{dx}(e^{2x} - 1) = 2e^{2x}$$ Taking the derivative of the denominator: $$\frac{d}{dx}(e^x - 1) = e^x$$ Now we have: $$\lim_{x \rightarrow 0} \frac{2e^{2x}}{e^x}$$

Step 4: Calculate the limit

Evaluating the limit, we see that the function is well-defined at $$x=0$$: $$\lim_{x \rightarrow 0} \frac{2e^{2x}}{e^x} = \frac{2e^{2(0)}}{e^0} = \frac{2(1)}{1} = 2$$ Thus, $$\lim_{x \rightarrow 0} f(x) = 2$$.

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a powerful mathematical tool used to find limits of functions that result in indeterminate forms, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). This rule is particularly useful when a straightforward evaluation of a limit is not possible due to these indeterminate forms.

Here's how it works: If you have two functions \(f(x)\) and \(g(x)\) and their limit as \(x\) approaches a particular value results in either a \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) form, then:

• You can take the derivative of each function separately.
• The limit of the ratio of their derivatives can then be evaluated.

Mathematically, if \(\lim_{x \to c} f(x) = \lim_{x \to c} g(x) = 0\) or \(\pm \infty\), then:\[ \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)} \] This approach simplifies the process because derivatives often transform the original functions into more manageable forms.

Limits

Limits are a fundamental concept in calculus used to understand the behavior of functions as they approach a specific point. Essentially, a limit asks what value a function "gets close to" as the input (or \(x\)-value) approaches some number.

Consider the problem where you need to find \(\lim_{x \to 0} f(x)\). Directly substituting \(x = 0\) might create an indeterminate form, like \(\frac{0}{0}\). In such cases, other methods (like factoring, simplifying, or using L'Hôpital's Rule) can be applied.

Key points about limits include:

• The limit exists only if the function approaches the same value from both sides of the point.
• Limits help define concepts like continuity, derivatives, and integrals.
• They are essential for handling situations where functions are not well-defined at certain points.

By understanding limits, you gain insight into the overall behavior of a function, particularly near any points of discontinuity.

Exponential Functions

Exponential functions are incredibly important in mathematics and science. They often take the form \(f(x) = a^{(bx+c)} + d\) where \(a\), \(b\), \(c\), and \(d\) are constants and the variable \(x\) appears in the exponent.

These functions are characterized by their unique rate of growth or decay, depending on the base \(a\). The basic exponential function \(f(x) = e^x\) is particularly significant because the base \(e\) is approximately equal to 2.71828. It offers a continuous rate of growth.

Key properties of exponential functions include:

• They are continuous for all real numbers.
• An exponential function grows faster than any polynomial function as \(x\) becomes large (in the case of growth).
• For \(e^x\), the derivative \(\frac{d}{dx}e^x = e^x\) reveals its property of self-derivation.

These properties make exponential functions highly applicable in various fields such as finance for compound interest calculations, biology for population models, and physics for radioactive decay.