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Chapter 3: Problem 6

Find the limit, if it exists. $$ \lim _{x \rightarrow \infty}\left(7+\frac{3}{x}\right) $$

Short Answer

Expert verified
The limit is 7.

Step by step solution

01

- Understanding the problem

We need to find the limit of the expression \(7 + \frac{3}{x} \) as \( x \) approaches infinity. This means we are interested in knowing what value the expression approaches when \( x \) becomes very large.
02

- Analyzing the term \( \frac{3}{x} \)

As \( x \) approaches infinity, the denominator of the term \( \frac{3}{x} \) becomes very large. Mathematically, this means \( \frac{3}{x} \) approaches 0 because the fraction \( \frac{3}{x} \) becomes smaller and smaller.
03

- Simplifying the expression

Given that \( \frac{3}{x} \) approaches 0, we can rewrite the expression: \[ 7 + \frac{3}{x} \to 7 + 0 \] This simplifies to \( 7 \).
04

- Stating the limit

Therefore, the limit of \( 7 + \frac{3}{x} \) as \( x \) approaches infinity is 7.

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Key Concepts

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

infinity in calculus
When we talk about infinity in calculus, we refer to a value that grows without bound. It鈥檚 essential to understand that infinity is not a number but rather a concept of something unbounded. This is particularly important when analyzing limits. For example, when we say that a variable, like x in the expression \(7 + \frac{3}{x}\), approaches infinity, we mean that x becomes very large. In practical terms, as x increases, \( \frac{3}{x} \) becomes smaller and smaller, approaching 0.
limits of functions
The limit of a function describes the behavior of that function as the input approaches a particular value. For instance, finding \( \lim_{x \to \infty} (7 + \frac{3}{x})\) means determining what value the function \( 7 + \frac{3}{x} \) approaches as x becomes very large. We break this down:
- Analyze the individual terms.
- Recognize that \( \frac{3}{x} \) becomes very small as x increases.
- Conclude that \( \frac{3}{x} \) approaches 0.
This leaves us with \ 7 + 0 = 7 \.
asymptotic behavior
Asymptotic behavior refers to the way a function behaves as the input grows very large or very small. In our example, the function \( 7 + \frac{3}{x} \) approaches a horizontal asymptote at y = 7 as x goes to infinity. An asymptote can be:
  • Horizontal: as shown in the example, where \ 7 + \frac{3}{x} \ levels out at y = 7.
  • Vertical: when the function grows unbounded at a specific x-value.
  • Oblique: takes the form of a slanted line.
This concept is critical for understanding how functions behave at extreme values of the input variable.

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