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Chapter 12: Problem 9

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) \\]

Short Answer

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

Step by step solution

01

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.
02

Limit Calculation

Calculate the limit by substituting the value that \( x \) is approaching into the function: \[\lim_{x\to \infty} \left(\dfrac{3}{x^2} + 1 \right) = \lim_{x\to \infty} \left(0 + 1 \right) \]
03

Solution Simplification

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

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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 (鈭�) 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鈥檚 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鈥攍ines 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.

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Most popular questions from this chapter

AVERAGE COST The cost function for a company to recycle \(x\) tons of material is given by \(C=1.25x + 10,500\), where \(C\) is measured in dollars. (a) Write a model for the average cost per ton of material recycled. (b) Find the average costs of recycling 100 tons of material and 1000 tons of material. (c) Determine the limit of the average cost function as \(x\) approaches infinity. Explain the meaning of the limit in the context of the problem.

In Exercises 29-36, complete the table using the function \( f(x) \), over the specified interval [a, b], to approximate the area of the region bounded by the graph of the \( y = f(x) \), the x-axis, and the vertical lines \(x=a\) and \(x=b\) and using the indicated number of rectangles. Then find the exact area as \( n \to \infty \). $$ f(x) = \frac{1}{2}x + 1 $$ Interval \( [-2, 2] \)

CONJECTURE Consider the functions \(f(x) = x^3\) and \(g(x) = x^3\). (a) Sketch the graphs of \(f\) and \(f'\) on the same set of coordinate axes. (b) Sketch the graphs of \(g\) and \(g'\) on the same set of coordinate axes. (c) Identify any pattern between the functions \(f\) and \(g\) and their respective derivatives. Use the pattern to make a conjecture about \(h'(x)\) if \(h(x) = x^n\), where \(n\) is an integer and \(n \geq 2\).

In Exercises 5-12, evaluate the sum using the summation formulas and properties. $$\displaystyle\sum_{i=1}^{60} 7$$

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} \dfrac{3x^2-4}{1-x^2} \\]
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