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Chapter 3: Problem 4

Find the limit, if it exists. $$ \lim _{x \rightarrow \infty} \frac{3 x+1}{4 x} $$

Short Answer

Expert verified
The limit is \frac{3}{4}

Step by step solution

01

Understand the given limit expression

The limit expression given is \( \lim _{ x \rightarrow \infty} \frac{3x+1}{4x} \). We need to find the value of this expression as \( x \) approaches infinity.
02

Simplify the fraction

Factor the term \( x \) out of both the numerator and the denominator: \[ \frac{3x+1}{4x} = \frac{3x}{4x} + \frac{1}{4x} = \frac{3}{4} + \frac{1}{4x} \]
03

Evaluate the simplified expression as \( x \) approaches infinity

As \( x \) approaches infinity, the term \(\frac{1}{4x}\) approaches 0 because the denominator grows very large. So, \(\frac{1}{4x} \rightarrow 0 \text{ as } x \rightarrow \infty)\).
04

Combine the results to find the limit

Using the fact that \(\frac{1}{4x}\) approaches 0, we have: \[ \lim _{ x \rightarrow \infty} \left( \frac{3}{4} + \frac{1}{4x} \right) = \frac{3}{4} + 0 = \frac{3}{4} \]

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Key Concepts

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

Limits at Infinity
When we talk about limits at infinity, we are interested in what happens to a function as the input value becomes very large, either in the positive or negative direction. This means, we want to know the behavior of the function as \( x \) approaches infinity or negative infinity. In the given problem, we are looking for the limit as \( x \rightarrow \infty \). For rational functions like \( f(x) = \frac{3x+1}{4x} \), we often see that some terms dominate as \( x \) grows large.
Simplifying Rational Expressions
Rational expressions are fractions involving polynomials. Simplifying these expressions can make finding limits much easier. In our exercise, we have \( \frac{3x + 1}{4x} \). To simplify, we factor out the common term \( x \) from both the numerator and the denominator. This gives us:
\[ \frac{3x+1}{4x} = \frac{3x}{4x} + \frac{1}{4x} \]
Notice that we now have terms that are easier to evaluate as \( x \) grows large.
Evaluating Limits
Once we have simplified our rational expression, we can now evaluate the limit. For our problem:
  • \( \frac{3x}{4x} = \frac{3}{4} \) because the \( x \) terms cancel out.
  • \( \frac{1}{4x} \) approaches 0 as \( x \rightarrow \infty \) because the denominator becomes very large.

So, combining these results:
\[ \lim_{x \rightarrow \infty} \left( \frac{3}{4} + \frac{1}{4x} \right) = \frac{3}{4} + 0 = \frac{3}{4} \]
This tells us that as \( x \) becomes very large, our original expression \( \frac{3x+1}{4x} \) approaches the value \( \frac{3}{4} \).

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