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Limit of (3x² + 2x) / (x² + 1) as x Approaches Infinity
Published on October 7, 2024
Calculus
Learn how to find the limit of the rational function (3x² + 2x) / (x² + 1) as x approaches infinity. The solution reveals a simple technique for evaluating limits at infinity and arrives at the answer: 3.
Question
Find the limit: lim(x→∞) (3x² + 2x) / (x² + 1)
Answer
Here’s how to find the limit step-by-step:
1. Identify the highest power of x in the expression.
Both the numerator and denominator have a highest power of x².
2. Divide each term in the numerator and denominator by x raised to the highest power.
Dividing each term by x²:
lim(x→∞) (3x² + 2x) / (x² + 1) = lim(x→∞) [(3x²/x²) + (2x/x²)] / [(x²/x²) + (1/x²)]
3. Simplify the expression.
lim(x→∞) [(3x²/x²) + (2x/x²)] / [(x²/x²) + (1/x²)] = lim(x→∞) (3 + 2/x) / (1 + 1/x²)
4. Evaluate the limit as x approaches infinity.
As x approaches infinity:
- 2/x approaches 0
- 1/x² approaches 0
Therefore, the limit becomes:
lim(x→∞) (3 + 2/x) / (1 + 1/x²) = (3 + 0) / (1 + 0) = 3/1
Answer:
lim(x→∞) (3x² + 2x) / (x² + 1) = 3