Limit of a Rational Function as x approaches Infinity: (3x^2 + 2x) / (x^2 + 1)

Published on November 5, 2025

Calculus

This problem involves finding the limit of a rational function as x approaches infinity. The solution involves dividing by the highest power of x in the denominator or by comparing the leading coefficients of the numerator and denominator.

Question

Find the limit: lim(x→∞) (3x² + 2x) / (x² + 1)

Answer

To find the limit of the given rational function as

approaches infinity, we can use the technique of dividing both the numerator and the denominator by the highest power of

in the denominator.

The given limit is:

The highest power of

in the denominator is

. So, we divide every term in the numerator and the denominator by

Simplify the terms:

Now, we evaluate the limit of each term as

:
As

(since the numerator is constant and the denominator goes to infinity)
(since the numerator is constant and the denominator goes to infinity)

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

(or

) is the ratio of the leading coefficients.

In this case, the numerator is

, and its leading coefficient is

.
The denominator is

, and its leading coefficient 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

Theorems

• Limit of a Rational Function (Equal Degrees)

For a rational function

, where

and

are polynomials, if the degree of

is equal to the degree of

, then the limit as

approaches infinity is the ratio of the leading coefficients of

and

. Specifically, if

and

, and

, then

• Limit Properties and Behavior at Infinity

When evaluating the limit of a function as

approaches infinity, if the function can be expressed as a sum or ratio of simpler functions, the limit of the combined function can often be found by taking the limit of each individual part. For example,

and

, provided the individual limits exist and the denominator limit is not zero. Also, for any constant

and any positive rational number

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