Problem 9 - Precalculus with Limits

Question

In Exercises 9-28, find the limit (if it exists). If the limit does not exist, explain why. Use a graphing utility to verify your result graphically. \\[\lim_{x\to \infty} \left(\dfrac{3}{x^2} + 1 \right) \\]

Step-by-Step Solution

Answer

The limit of the function \( \frac{3}{x^2} + 1 \) as \( x \) approaches infinity is 1.

Step 1: Function Analysis

Analyze the function \( \frac{3}{x^2} + 1 \). As \( x \) grows larger, \(\frac{1}{x^2}\) approaches 0. Therefore, \( \frac{3}{x^2} \) also converges towards 0.

Step 2: Limit Calculation

Calculate the limit by substituting the value that \( x \) is approaching into the function: \[\lim_{x\to \infty} \left(\dfrac{3}{x^2} + 1 \right) = \lim_{x\to \infty} \left(0 + 1 \right) \]

Step 3: Solution Simplification

Simplify the expression for the final solution, which gives us 1.

Key Concepts

Infinite Limits

Understanding infinite limits is about seeing how a function behaves as the input values grow larger and larger. For the function \( \frac{3}{x^2} + 1 \), consider what happens as \( x \) approaches infinity. The term \( \frac{3}{x^2} \) becomes smaller because we divide by an ever-increasing number. Essentially, it gets so close to zero that it becomes negligible. So, as \( x \) moves towards infinity, the term \( \frac{3}{x^2} \) approaches zero.

Simplifying this, the expression \( \frac{3}{x^2} + 1 \) behaves like \( 0 + 1 \) as \( x \to \infty \), resulting in the limit being 1.

Remember:

• The infinity symbol (∞) represents an unbounded large number.
• An infinite limit often illustrates a direction rather than a finite value, unless simplified to a constant.

Graphical Verification

Using a graphing utility can help to visualize functions and verify the behavior of limits. When we graph the function \( \frac{3}{x^2} + 1 \), the effects are clear because you can see:

• The graph approaches the horizontal line \( y = 1 \) as \( x \) increases.
• This graphical behavior confirms the calculated limit \( =1 \).

Graphical verification supports our mathematical solution by showing visually that the function's output stabilizes around 1 when \( x \) is very large. It’s a great tool to build intuition around how functions work at extremes.

Verify with Steps:

• Start by plotting \( y = \frac{3}{x^2} + 1 \) with increasing \( x \)-values.
• Look at the function's approach towards the horizontal asymptote, \( y = 1 \).
• Notice how the graph flattens out, confirming the limit \( \to 1 \) as \( x \to \infty \).

Rational Functions

Rational functions are expressions that involve ratios of polynomials, like \( \frac{3}{x^2} + 1 \). These functions can have complex behaviors, especially as their inputs grow larger or smaller.

Key characteristics:

• They often feature asymptotes—lines that the graph approaches but never quite touches, providing a visual representation of limits.
• The behavior of a rational function is largely determined by the degree of its polynomial components. For example, \( \frac{3}{x^2} \) diminishes to zero at large \( x \).

When considering infinite limits in rational functions, always look at the highest powers in the numerator and denominator. Here, the term \( \frac{3}{x^2} \) is overshadowed by the constant 1 as \( x \to \infty \), clarifying why the limit simplifies to 1.