Question
How to find limit of an infinity with example
Answers
GPT-4o mini
Finding limits involving infinity often requires evaluating the behavior of functions as they approach infinity or as the variables approach infinity. Here are a few common scenarios and methods to find such limits, along with an example.
### Steps to Find Limits Involving Infinity
1. **Direct Substitution**: If the function is well-defined at infinity, you can directly substitute ∞ into the function. However, in most cases, direct substitution will lead to an indeterminate form.
2. **Simplification**: Simplify the function, if possible. This might involve factoring, expanding, or combining fractions.
3. **L'Hôpital's Rule**: If you encounter an indeterminate form like \( \frac{\infty}{\infty} \) or \( \frac{0}{0} \), apply L'Hôpital's Rule. This entails taking the derivative of the numerator and the derivative of the denominator separately and evaluating the limit again.
4. **Dominant Terms**: As \( x \to \infty \), determine the dominant term in the function that will dictate the behavior of the limit. Often, the highest power of \( x \) in the polynomial will determine the behavior.
5. **Limit Definitions**: Use definitions of limits at infinity, if applicable for your function (like approaching zero or becoming unbounded).
### Example
Let's find the limit of the function:
\[
\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{5x^2 + 4}
\]
**Step 1: Determine the leading terms**
- As \( x \) approaches infinity, the leading terms of both the numerator and the denominator dominate the behavior of the function:
- Numerator: \( 2x^2 \)
- Denominator: \( 5x^2 \)
**Step 2: Simplify the function**
- Dividing both the numerator and the denominator by \( x^2 \):
\[
\lim_{x \to \infty} \frac{2 + \frac{3}{x} + \frac{1}{x^2}}{5 + \frac{4}{x^2}}
\]
**Step 3: Evaluate the limit**
- Now as \( x \to \infty \), the fractions \( \frac{3}{x} \) and \( \frac{1}{x^2} \) both approach 0, and so does \( \frac{4}{x^2} \):
\[
\lim_{x \to \infty} \frac{2 + 0 + 0}{5 + 0} = \frac{2}{5}
\]
Thus,
\[
\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{5x^2 + 4} = \frac{2}{5}
\]
### Conclusion
In this example, the limit as \( x \) approaches infinity of the function \( \frac{2x^2 + 3x + 1}{5x^2 + 4} \) is \( \frac{2}{5} \). This shows how understanding the dominant terms in the polynomial can quickly lead to finding limits at infinity.
### Steps to Find Limits Involving Infinity
1. **Direct Substitution**: If the function is well-defined at infinity, you can directly substitute ∞ into the function. However, in most cases, direct substitution will lead to an indeterminate form.
2. **Simplification**: Simplify the function, if possible. This might involve factoring, expanding, or combining fractions.
3. **L'Hôpital's Rule**: If you encounter an indeterminate form like \( \frac{\infty}{\infty} \) or \( \frac{0}{0} \), apply L'Hôpital's Rule. This entails taking the derivative of the numerator and the derivative of the denominator separately and evaluating the limit again.
4. **Dominant Terms**: As \( x \to \infty \), determine the dominant term in the function that will dictate the behavior of the limit. Often, the highest power of \( x \) in the polynomial will determine the behavior.
5. **Limit Definitions**: Use definitions of limits at infinity, if applicable for your function (like approaching zero or becoming unbounded).
### Example
Let's find the limit of the function:
\[
\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{5x^2 + 4}
\]
**Step 1: Determine the leading terms**
- As \( x \) approaches infinity, the leading terms of both the numerator and the denominator dominate the behavior of the function:
- Numerator: \( 2x^2 \)
- Denominator: \( 5x^2 \)
**Step 2: Simplify the function**
- Dividing both the numerator and the denominator by \( x^2 \):
\[
\lim_{x \to \infty} \frac{2 + \frac{3}{x} + \frac{1}{x^2}}{5 + \frac{4}{x^2}}
\]
**Step 3: Evaluate the limit**
- Now as \( x \to \infty \), the fractions \( \frac{3}{x} \) and \( \frac{1}{x^2} \) both approach 0, and so does \( \frac{4}{x^2} \):
\[
\lim_{x \to \infty} \frac{2 + 0 + 0}{5 + 0} = \frac{2}{5}
\]
Thus,
\[
\lim_{x \to \infty} \frac{2x^2 + 3x + 1}{5x^2 + 4} = \frac{2}{5}
\]
### Conclusion
In this example, the limit as \( x \) approaches infinity of the function \( \frac{2x^2 + 3x + 1}{5x^2 + 4} \) is \( \frac{2}{5} \). This shows how understanding the dominant terms in the polynomial can quickly lead to finding limits at infinity.