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Chapter 1: Problem 57

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{5 x+3}{3 x-2}$$

Short Answer

Expert verified
The limit is \(\frac{5}{3}\)

Step by step solution

01

Identify the given limit

The given limit is \[\lim_{x \rightarrow \infty} \frac{5x+3}{3x-2}\]
02

Determine the degrees of the numerator and the denominator

The degree of the numerator \(5x + 3\) and the denominator \(3x - 2\) are both 1 since the highest power of \(x\) in both expressions is 1.
03

Divide through by the highest power of \(x\)

To simplify, divide both the numerator and the denominator by \(x\):\[\lim_{x \rightarrow \infty} \frac{5x+3}{3x-2} = \lim_{x \rightarrow \infty} \frac{\frac{5x}{x} + \frac{3}{x}}{\frac{3x}{x} - \frac{2}{x}} = \lim_{x \rightarrow \infty} \frac{5 + \frac{3}{x}}{3 - \frac{2}{x}}\]
04

Evaluate the limit as \(x\) approaches infinity

As \(x\) approaches infinity, the terms \(\frac{3}{x}\) and \(\frac{2}{x}\) approach 0. So, the limit simplifies to:\[\lim_{x \rightarrow \infty} \frac{5 + 0}{3 - 0} = \frac{5}{3}\]

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Key Concepts

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

Infinity Limits
Infinity limits help us understand the behavior of functions as variables grow extremely large. In this exercise, we evaluate the limit of a rational function as \(x\) approaches infinity.
Singular terms like \(\frac{3}{x}\) and \(\frac{2}{x}\) shrink to zero since dividing any number by an extremely large number makes the result very small.
As a result, they vanish in the presence of much larger terms.
This simplification is key when computing infinity limits for rational functions.
Here, as \( x \rightarrow \infty \), these fractions approach 0, simplifying our limit calculation.
Rational Functions
Rational functions are functions in the form \(f(x) = \frac{P(x)}{Q(x)}\), where \(P(x)\) and \(Q(x)\) are polynomials.
In our case, \( P(x) = 5x + 3 \) and \( Q(x) = 3x - 2 \).
These functions can exhibit complex behaviors, but they simplify significantly when we deal with limits at infinity.
By dividing both numerator and denominator by the highest power of \( x \) present, we obtain:
\lim_{x \rightarrow \infty} \frac{5x + 3}{3x - 2} = \lim_{x \rightarrow \infty} \frac{5 + \frac{3}{x}}{3 - \frac{2}{x}}\
This form reveals that as \( x \rightarrow \infty \), the fractional components approach 0, making it easier to calculate the overall limit.
Asymptotic Behavior
Asymptotic behavior describes how functions behave as inputs become very large or very small.
In this problem, we focus on the behavior as \( x \rightarrow \infty \).
The simplified form \frac{5 + \frac{3}{x}}{3 - \frac{2}{x}}\ shows that as \( x \) increases, the \( \frac{3}{x} \) and \( \frac{2}{x} \) terms become negligible.
This reduces our rational function to \frac{5}{3}\.
Thus, as \( x \) gets large, the function value approaches the constant ratio of the coefficients of the highest degree terms in the numerator and denominator, highlighting the importance of leading terms in determining asymptotic behavior.

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Most popular questions from this chapter

Determine which of the following limits exist. Compute the limits that exist. Use the limit definition of the derivative to show that if \(f(x)=m x+b,\) then \(f^{\prime}(x)=m.\)

Let \(f(x)\) be the value in dollars of one share of a company \(x\) days since the company went public. (a) Interpret the statements \(f(100)=16\) and \(f^{\prime}(100)=.25.\) (b) Estimate the value of one share on the 101 st day since the company went public.

Use limits to compute \(f^{\prime}(x) .\) $$f(x)=3 x+1$$

Compute the following limits. $$\lim _{x \rightarrow \infty} \frac{1}{x^{2}}$$

Determine which of the following limits exist. Compute the limits that exist. $$\lim _{x \rightarrow 8} \frac{x^{2}+64}{x-8}$$
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