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Chapter 1: Problem 63

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 limit expression

The task is to compute the limit as x approaches infinity for the expression \(\frac{5x+3}{3x-2}\).
02

Divide numerator and denominator by x

To simplify the expression, divide both the numerator and the denominator by x, the highest power of x present in the denominator. This gives \(\frac{\frac{5x+3}{x}}{\frac{3x-2}{x}}\).
03

Simplify the fraction

Simplify the fraction by performing the divisions: \(\frac{\frac{5x}{x}+\frac{3}{x}}{\frac{3x}{x}-\frac{2}{x}} = \frac{5+\frac{3}{x}}{3-\frac{2}{x}}\).
04

Apply the limit

As x approaches infinity, the terms \( \frac{3}{x} \) and \( \frac{2}{x} \) approach 0. Therefore, the expression simplifies to \(\frac{5+0}{3-0} = \frac{5}{3}\).

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Key Concepts

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

limit at infinity
When we talk about the 'limit at infinity' in calculus, we're examining the behavior of a function as the variable approaches infinity. In simpler terms, we're seeing what happens to the function's value when x gets really, really large. For example, consider the limit problem: \ \(\lim_{x \rightarrow \infty} \frac{5x+3}{3x-2}\). As x becomes very large, the values of 5x+3 and 3x-2 also grow. Understanding how these terms behave helps us find the limit.
simplifying fractions
Simplifying fractions is a crucial step in solving limit problems. It involves breaking down the function into simpler parts. In our example, \ \(\frac{5x+3}{3x-2}\), we divided both numerator and denominator by x, the highest power in the fraction: \ \(\frac{\frac{5x+3}{x}}{\frac{3x-2}{x}} \). By performing these divisions, we get: \ \(\frac{5+\frac{3}{x}}{3-\frac{2}{x}}\), which is much easier to handle. When x approaches infinity, terms with x in the denominator approach zero, further simplifying our function.
rational functions
Rational functions are expressions where both the numerator and denominator are polynomials. For example, \ \(\frac{5x+3}{3x-2}\) is a rational function. Understanding the properties of these functions helps in determining their limits. A key property to note here is that as x becomes very large, lower-degree terms (like the constants +3 and -2) become insignificant compared to the higher-degree terms (those with x). This insight allows us to focus on simplifying the higher-degree terms.
calculus problem solving
Solving calculus problems often involves breaking the problem into smaller, manageable steps. For example, consider our problem: \ \(\lim_{x \rightarrow \infty} \frac{5x+3}{3x-2}\). The steps are: \ \
    \ \
  • Identify the limit expression.\ \
    • \ \
  • Divide numerator and denominator by the highest power of x present.\ \
    • \ \
  • Simplify the fraction.\ \
    • \ \
  • Apply the limit.\ \
    • \ \
By following these steps, we can simplify complex problems into easy-to-follow actions, making calculus more approachable.

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Most popular questions from this chapter

Use limits to compute \(f^{\prime}(x)\). [Hint: In Exercises \(45-48\), use the rationalization trick of Example \(8 .]\) \(f(x)=\frac{1}{x^{2}+1}\)

Determine whether each of the following functions is continuous and/or differentiable at \(x=1\). \(f(x)=\left\\{\begin{array}{ll}2 x-1 & \text { for } 0 \leq x \leq 1 \\ 1 & \text { for } 1

Compute the difference quotient $$ \frac{f(x+h)-f(x)}{h} . $$ Simplify your answer as much as possible. \(f(x)=x^{2}-7\)

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. \(f^{\prime}(0)\), where \(f(x)=10^{1+x}\)

Use a derivative routine to obtain the value of the derivative. Give the value to 5 decimal places. \(f^{\prime}(1)\), where \(f(x)=\sqrt{1+x^{2}}\)
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