Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to find the asymptote\((s)\) of each function. $$\lim _{x \rightarrow \infty}(\sqrt{x^{2}+x}-x)$$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \).

Step 1: Simplify in the Limit

Start by simplifying the expression \( \sqrt{x^2 + x} - x \). Observe that for large \( x \), both \( \sqrt{x^2 + x} \) and \( x \) tend towards infinity. Simplify by factoring out the highest power of \( x \) from under the square root: \( \sqrt{x^2 + x} = \sqrt{x^2(1 + \frac{1}{x})} = x\sqrt{1 + \frac{1}{x}} \). This makes the expression \( x(\sqrt{1 + \frac{1}{x}}) - x \).

Step 2: Further Simplify the Expression

Factor out x in the expression to make it \( x(\sqrt{1 + \frac{1}{x}} - 1) \).

Step 3: Expand Square Root Expression

Now, expand \( \sqrt{1 + \frac{1}{x}} \) using the binomial approximation for small values of \( \frac{1}{x} \). This results in \( \sqrt{1 + \frac{1}{x}} \approx 1 + \frac{1}{2x} \). Substitute back to get \( x(1 + \frac{1}{2x} - 1) = x\frac{1}{2x} = \frac{1}{2}. \)

Step 4: Compute the Limit

Hence, as \( x \to \infty \), the expression \( \sqrt{x^2 + x} - x \) approaches \( \frac{1}{2} \).

Key Concepts

Understanding Limits

Limits are a fundamental concept in calculus, commonly used to determine the behavior of a function as the input approaches a certain value. When we say "the limit of a function as x approaches a value," we are interested in what value the function is getting closer to.
For example, if we have a function \( f(x) = x^2 \), and we want to find out what happens as \( x \) approaches 3, we use the notation \( \lim_{x \to 3} f(x) = 3^2 = 9 \).
In our exercise, we are examining the limit as \( x \) approaches infinity. This is another way of saying, "What happens to the function when \( x \) gets extremely large?"
The limit helps to describe the end behavior of functions and is vital when identifying horizontal asymptotes.

Exploring Infinite Limits

Infinite limits occur when the values of a function increase or decrease without bound as the input approaches a certain value. When this happens, we use infinity (\(\infty\)) or negative infinity (\(-\infty\)) to describe the behavior.
Consider a simple function, \( f(x) = \frac{1}{x} \). As \( x \to 0 \) from the positive side, the function approaches positive infinity \( \lim_{x \to 0^+} \frac{1}{x} = \infty \). Conversely, as \( x \to 0 \) from the negative side, \( \lim_{x \to 0^-} \frac{1}{x} = -\infty \).
However, in our original exercise involving \( \sqrt{x^2+x} - x \), the limit is evaluated as \( x \to \infty \), and the simplification shows that the expression remains finite, reaching a specific value instead of going to infinity.

Understanding Binomial Approximation

Binomial approximation is a mathematical technique used to simplify expressions involving roots or powers, especially when the input becomes very large or small.
In our case, we approximated \( \sqrt{1 + \frac{1}{x}} \) using binomial expansion, leading to \( \approx 1 + \frac{1}{2x} \) for large values of \( x \).
This is based on the binomial theorem which in its simplest form for such cases states that for very small \( y \), \( (1 + y)^n \approx 1 + ny \), where \( n \) here is \( \frac{1}{2} \).
This approximation allows us to compute limits more easily and is instrumental for finding precise answers when working with complex expressions.

Diving into Rational Functions

Rational functions are expressions formed by the ratio of two polynomials, like \( \frac{p(x)}{q(x)} \). They often have asymptotes which can be horizontal, vertical, or oblique.
For the exercise provided, the expression \( \sqrt{x^2 + x} - x \) initially appears as a subtraction rather than a division, yet rational functions play a role in its simplification.
By breaking down \( \sqrt{x^2 + x} \) as \( x\sqrt{1 + \frac{1}{x}} \), we conceptually create a scenario similar to a rational function, where the behavior at infinity can be analyzed through limits.
Understanding how these functions behave at extreme values helps us identify asymptotes, as demonstrated by the expression tending toward \( \frac{1}{2} \) instead of increasing or decreasing indefinitely.