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Limit of Rational Function as x Approaches Infinity: (3x^2 + 2x) / (x^2 + 1)
Published on August 31, 2025
Calculus
Find the limit of the rational function (3x^2 + 2x) / (x^2 + 1) as x approaches infinity by dividing by the highest power of x in the denominator or by comparing leading coefficients.
Question
Find the limit: lim(x→∞) (3x² + 2x) / (x² + 1)
Answer
This is a math problem asking to find the limit of a rational function as x approaches infinity.
We need to find the limit:
To evaluate this limit, we can divide both the numerator and the denominator by the highest power of x in the denominator, which is
Step 1: Divide the numerator by
Step 2: Divide the denominator by
Step 3: Rewrite the expression with the divided terms:
Step 4: Apply the limit as
So,
So,
Step 5: Substitute these limits back into the expression:
Alternatively, for rational functions where the degree of the numerator is equal to the degree of the denominator, the limit as
In this problem, the numerator is
The denominator is
Since the degree of the numerator (2) is equal to the degree of the denominator (2), the limit is the ratio of the leading coefficients.
The denominator is
Since the degree of the numerator (2) is equal to the degree of the denominator (2), the limit is the ratio of the leading coefficients.
The final answer is