Find the Asymptotes (8e^x)/(e^x-7)

Solution:

Step 1:

Identify the values of $x$ where the function $\frac{8 e^{x}}{e^{x} - 7}$ is not defined. This occurs when the denominator equals zero, which happens at $x = \ln(7)$.

Step 2:

To determine the horizontal asymptote as $x \to \infty$, calculate the limit $\lim_{x \to \infty} \frac{8 e^{x}}{e^{x} - 7}$.

Step 2.1:

Extract the constant $8$ from the limit expression: $8 \lim_{x \to \infty} \frac{e^{x}}{e^{x} - 7}$.

Step 2.2:

Invoke L'Hospital's Rule to evaluate the limit.

Step 2.2.1:

Compute the limits of the numerator and denominator separately: $8 \frac{\lim_{x \to \infty} e^{x}}{\lim_{x \to \infty} e^{x} - 7}$.

Step 2.2.2:

Recognize that as $x \to \infty$, $e^{x} \to \infty$: $8 \frac{\infty}{\lim_{x \to \infty} e^{x} - 7}$.

Step 2.2.3:

Apply the Sum of Limits Rule: $8 \frac{\infty}{\lim_{x \to \infty} e^{x} - \lim_{x \to \infty} 7}$.

Step 2.2.4:

Since $e^{x} \to \infty$ as $x \to \infty$: $8 \frac{\infty}{\infty - \lim_{x \to \infty} 7}$.

Step 2.2.5:

Evaluate the limit of the constant $7$: $8 \frac{\infty}{\infty - 7}$.

Step 2.2.6:

Simplify the expression: $8 \frac{\infty}{\infty}$.

Step 2.2.7:

Recognize that $\frac{\infty}{\infty}$ is an indeterminate form and apply L'Hospital's Rule again: $\lim_{x \to \infty} \frac{e^{x}}{e^{x} - 7} = \lim_{x \to \infty} \frac{d}{dx} e^{x} / \frac{d}{dx} (e^{x} - 7)$.

Step 2.3:

Differentiate the numerator and denominator.

Step 2.3.1:

The derivative of $e^{x}$ is $e^{x}$, and the derivative of $e^{x} - 7$ is $e^{x}$: $8 \lim_{x \to \infty} \frac{e^{x}}{e^{x}}$.

Step 2.3.2:

Cancel the common factors of $e^{x}$: $8 \lim_{x \to \infty} 1$.

Step 2.3.3:

Evaluate the limit of the constant $1$: $8 \cdot 1 = 8$.

Step 3:

To find the horizontal asymptote as $x \to -\infty$, calculate the limit $\lim_{x \to -\infty} \frac{8 e^{x}}{e^{x} - 7}$.

Step 3.1:

Evaluate the limit: $8 \lim_{x \to -\infty} \frac{e^{x}}{e^{x} - 7}$.

Step 3.2:

As $x \to -\infty$, $e^{x} \to 0$: $8 \frac{0}{\lim_{x \to -\infty} e^{x} - 7}$.

Step 3.3:

Apply the Sum of Limits Rule: $8 \frac{0}{0 - \lim_{x \to -\infty} 7}$.

Step 3.4:

Evaluate the limit of the constant $7$: $8 \frac{0}{0 - 7}$.

Step 3.5:

Simplify the expression: $8 \cdot 0 = 0$.

Step 4:

Combine the results to list the horizontal asymptotes: $y = 8$ and $y = 0$.

Step 5:

There is no slant (oblique) asymptote because the degree of the numerator is not greater than the degree of the denominator.

Step 6:

Compile the complete set of asymptotes:

• Vertical Asymptote: $x = \ln(7)$
• Horizontal Asymptotes: $y = 8$ and $y = 0$
• No Oblique Asymptotes

Knowledge Notes:

1. Asymptotes: An asymptote is a line that a graph of a function approaches but never touches. There are three types of asymptotes: vertical, horizontal, and oblique (slant).

2. Vertical Asymptotes: These occur when the function approaches infinity or negative infinity as $x$ approaches a certain value. They can be found by setting the denominator equal to zero and solving for $x$.

3. Horizontal Asymptotes: These occur when the value of the function approaches a constant as $x$ approaches infinity or negative infinity. They can be found by evaluating the limit of the function as $x$ goes to infinity or negative infinity.

4. L'Hospital's Rule: This is a method to evaluate limits of indeterminate forms such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$. The rule states that if the limits of the numerator and denominator are both zero or both infinity, the limit of the quotient is the limit of the quotient of their derivatives.

5. Limits and Continuity: The concept of limits is fundamental to calculus and mathematical analysis and involves finding the value that a function approaches as the input approaches some value. Continuity, on the other hand, means that the function is defined at that point, and the limit at that point is equal to the function's value.

6. Exponential Functions: These are functions of the form $f(x) = a^x$, where $a$ is a positive constant. The base $e$ is the natural exponential function, where $e$ is an irrational and transcendental number approximately equal to 2.71828.

7. Sum of Limits Rule: This rule states that the limit of a sum is equal to the sum of the limits, provided that the limits exist.

8. Derivative of Exponential Functions: The derivative of $e^x$ with respect to $x$ is $e^x$. This property makes exponential functions particularly easy to differentiate and integrate.