What is the limit of 1/x as x approaches infinity?
Understand the Problem
The question is asking for the limit of the function 1/x as the variable x approaches infinity. This is a common limit in calculus and involves understanding the behavior of the function as x gets larger and larger.
Answer
$0$
Answer for screen readers
The limit is $0$.
Steps to Solve
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Identify the function and the variable We are working with the function $f(x) = \frac{1}{x}$ and looking at the limit as $x$ approaches infinity.
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Analyze the function as x increases As $x$ gets larger (i.e., moves towards infinity), we can think about the value of $f(x)$. When substituting larger and larger values for $x$, the function can be described as: $$ f(x) = \frac{1}{x} $$
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Compute the limit To find the limit, we evaluate: $$ \lim_{x \to \infty} \frac{1}{x} $$ As $x$ increases (for example, $10$, $100$, $1000$), the value of $\frac{1}{x}$ becomes smaller and approaches 0.
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Conclusion about the limit Thus, we establish that: $$ \lim_{x \to \infty} \frac{1}{x} = 0 $$
The limit is $0$.
More Information
This limit illustrates a fundamental concept in calculus, showing how a fractional function behaves as its denominator grows infinitely large. It indicates that as we divide a constant by an increasingly larger number, the result approaches zero.
Tips
- Assuming the limit approaches a constant other than zero. Remember, the behavior of the fraction $\frac{1}{x}$ means it gets indefinitely small as $x$ increases.
- Misinterpreting the limit notation; remember that $\lim_{x \to \infty}$ focuses on what happens to the function as $x$ becomes larger without bound.
