Problem 6 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(1-8,\) use graphs and tables to find (a) \(\lim _{x \rightarrow \infty} f(x)\) and (b) \(\lim _{x \rightarrow-\infty} f(x)\) (c) Identify all horizontal asymptotes. $$f(x)=\frac{2 x-1}{|x|-3}$$

Step-by-Step Solution

Answer

The limit as x approaches positive infinity is 2 and the limit as x approaches negative infinity is -2. The function f(x) has two horizontal asymptotes: y = 2 and y = -2.

Step 1: Analyze the function and simplify

Observe the function \(f(x) = \frac{2x-1}{|x|-3}\). Due to the absolute value, this signifies that the function takes different forms when x < 0 and x > 0. Thus, it can be split into two functions: \(f_1(x) = \frac{2x-1}{x-3}\) when x > 0 and \(f_2(x) = \frac{2x-1}{-x-3}\) when x < 0. This will help to calculate the limits when x approaches positive and negative infinity individually.

Step 2: Determine the limits when x approaches positive infinity

To calculate \( \lim_{x \to \infty} f_1(x)\), ignore the constants in numerator and denominator and consider the leading terms. Certainly, the limit will be the ratio of coefficients of highest degree terms. Hence, \( \lim_{x \to \infty} f_1(x) = \frac{2}{1} = 2.\)

Step 3: Determine the limits when x approaches negative infinity

To calculate \( \lim_{x \to -\infty} f_2(x) \), ignore the constants and consider the leading terms. Hence, \( \lim_{x \to -\infty} f_2(x) = \frac{2}{-1} = -2.\)

Step 4: Identify all horizontal asymptotes

Horizontal asymptotes for the function f(x) can be found by looking at the limits we’ve calculated. There are two horizontal asymptotes: y = 2 as x approaches positive infinity and y = -2 as x approaches negative infinity.

Key Concepts

Horizontal Asymptotes

Horizontal Asymptotes are lines that a graph approaches but never actually reaches as the input values (x) go to positive or negative infinity. They give us a simple idea about the behavior of a function at the 'ends' or 'tails' of the graph. In this exercise, we learned that the function \[f(x) = \frac{2x-1}{|x|-3}\] transforms into two different forms, one for when x is greater than zero and another for when x is less than zero. These transformations reveal the horizontal asymptotes at y=2 and y=-2.Being aware of such asymptotes helps in predicting the end behavior of functions without having to calculate every detail. For rational functions like the one in this exercise, it's especially critical and stems directly from the limits at positive and negative infinity.

Infinite Limits

Understanding Infinite Limits is key when assessing how functions behave as they reach large magnitudes either positively or negatively. Specifically, infinite limits tell us about the direction in which a function extends when x heads towards infinity or negative infinity.In the exercise, we identified the limits of \[f_1(x) = \frac{2x-1}{x-3}\] when x > 0 and \[f_2(x) = \frac{2x-1}{-x-3}\] when x < 0:

• As x approaches positive infinity for \(f_1(x)\), the ratio \( \lim_{x \to \infty} \frac{2}{1} = 2 \) indicates that the function flattens to approach y=2.
• As x approaches negative infinity for \(f_2(x)\), the ratio \( \lim_{x \to -\infty} \frac{2}{-1} = -2 \) shows that the function flattens to approach y=-2.

For other functions, procedures may vary but understanding the large-x behavior of a function through its leading coefficients helps graph it effectively.

Rational Functions

Rational Functions are functions that are expressed as the ratio of two polynomials. In simplest terms, they involve fractions, such as the exercise's \[f(x) = \frac{2x-1}{|x|-3}\].They are invaluable in calculus because they often model real-world scenarios – from physics to economics. A key feature of rational functions is that their limits at infinity often boil down to comparing the degrees and leading coefficients of the numerator and the denominator.Some handy facts about rational functions are:

• The horizontal asymptote can often be found by dividing the leading coefficients if the degrees of the numerator and denominator are the same.
• If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y=0.
• If the degree of the numerator is greater than that of the denominator, there is no horizontal asymptote, and the graph goes to either positive or negative infinity.

Understanding these basics helps in sketching graphs and predicting behavior quickly for varying inputs.