Problem 59 - Essential Calculus

Question

In Exercises 59 and \(60,\) (a) explain why L'Hôpital's Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\)

Step-by-Step Solution

Answer

The limit as \(x\) goes to infinity of \( \frac{x}{\sqrt{x^{2}+1}}\) is 1.

Step 1: Rewrite the Limit

Rewrite the limit \(\lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\) by factoring out \(x\) from the radical in the denominator.

Step 2: Simplify the Limit

When you factor out \(x\) from the root, you get the expression \( \lim_{x \rightarrow \infty} \frac{x}{x\sqrt{1+\frac{1}{x^{2}}}}\). After simplifying, you get \( \lim_{x \rightarrow \infty} \frac{1}{\sqrt{1+\frac{1}{x^2}}}\)

Step 3: Evaluate the Limit

As \(x\) goes to infinity, the expression \(\frac{1}{x^{2}}\) tends to 0, which simplifies the limit to \( \lim_{x \rightarrow \infty} \frac{1}{\sqrt{1}}\). Hence, the limit of the given function as \(x\) goes to infinity is 1.

Step 4: Graph the Function

Graph the function \( \frac{x}{\sqrt{x^{2}+1}}\) and observe its behaviour as \(x\) approaches to infinity, which indeed confirms the calculated limit.

Key Concepts

Limits at Infinity

When we talk about limits at infinity, we are essentially looking at the behavior of a function as the input variable grows larger and larger. Imagine plugging bigger and bigger numbers into your function. You want to find out what the output tends towards as these numbers skyrocket. In mathematical notation, this is expressed as \( x \to \infty \), translating to "x approaches infinity."

In essence:

• As the variable (typically x) increases, we're interested in the end behavior of the function.
• We endeavor to find what output value the function will eventually settle to as x gets very large.
• This behavior helps in identifying horizontal asymptotes in functions, guiding us to understand potential boundaries for the output values.

For the function \( \frac{x}{\sqrt{x^{2}+1}} \), as \( x \to \infty \), we simplify the function essentially to \( \frac{1}{\sqrt{1}} = 1 \). This shows us that no matter how large x becomes, our function's output nears this finite value of 1.

Analytical Techniques for Limits

Analytical techniques for limits involve using mathematical manipulations and algebraic steps to find the limit of a function without having to graph it. This can be very handy, as it provides a straightforward calculation to verify the behavior of functions at extreme values.

For example, with L'Hôpital's rule🔗 used for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), one may be tempted to use it here. However, our function does not yield an indeterminate form when you substitute larger values of x. Therefore, L'Hôpital's Rule doesn't apply, and we turn to other algebraic techniques.

By manipulating the function \( \lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}} \):

• Factor x from the square root in the denominator resulting in \( x\sqrt{1+\frac{1}{x^{2}}} \).
• Cancel out common terms to simplify it to \( \frac{1}{\sqrt{1+\frac{1}{x^{2}}}} \).
• As \( x \to \infty \), \( \frac{1}{x^{2}} \to 0 \), leaving the expression \( \frac{1}{\sqrt{1}} \) which simplifies to 1.

Through these steps, we analytically verify that the limit as x approaches infinity is indeed 1.

Graphical Approximation of Limits

Graphical approximation provides a visual insight into what happens to a function as x increases towards infinity. It’s an excellent way to check our analytical results by observing graphical behavior.

Using a graphing utility for the function \( \frac{x}{\sqrt{x^{2}+1}} \):

• Plot the function on your graphing calculator or software.
• Observe the curve as x grows larger and moves towards the right on the x-axis.
• You'll typically notice the curve approaching a horizontal line, suggesting it is trending towards a particular value.

In this case, the curve approaches the horizontal line, y = 1, visually confirming that the limit of our function as \( x \to \infty \) is 1. A graphical approach not only corroborates our analytical work but also provides an intuitive grasp of the function's end behavior.