in a limit if the numerator approaches a constant and the denominator approaches infinity, what is the limit 

If in a limit, the numerator approaches a constant \( c \) (where \( c \neq 0 \)) and the denominator approaches infinity, the limit will approach 0. This is because dividing a constant by an increasingly large number results in a value that gets closer and closer to zero.

Mathematically, this can be expressed as:

\[
\lim_{x \to \infty} \frac{c}{f(x)} = 0
\]

where \( c \) is a constant and \( f(x) \to \infty \) as \( x \to \infty \).

### Example:
If \( \lim_{x \to \infty} \frac{5}{x} \), the numerator (5) is a constant and the denominator grows without bound. Therefore, the limit is:

\[
\lim_{x \to \infty} \frac{5}{x} = 0
\]

### Final Answer:
When the numerator approaches a constant and the denominator approaches infinity, the limit is 0.

Would you like more details or have any questions?

Here are 8 related questions you might consider:
1. What happens if the numerator approaches 0 and the denominator approaches infinity?
2. How do you evaluate limits where both the numerator and denominator approach infinity?
3. What is the limit if both the numerator and denominator approach zero?
4. How do you apply L'Hopital's rule in cases like this?
5. What is the limit if the numerator approaches a constant and the denominator approaches a negative infinity?
6. How do you determine horizontal asymptotes using limits?
7. How do limits behave when both the numerator and denominator grow exponentially?
8. How do limits behave when the numerator approaches infinity and the denominator approaches a constant?

**Tip:** When analyzing limits, always carefully examine the behavior of both the numerator and the denominator as the variable approaches infinity or another critical value.

Explore the limit of a constant numerator and an infinity denominator in mathematics. Learn how dividing a constant by an increasingly large number approaches zero. Examples and detailed explanations included.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation