Take the limit of the numerator and the limit of the denominator.

The limit at infinity of a polynomial whose leading coefficient is positive is infinity.

As approaches for radicals, the value goes to .

Infinity divided by infinity is undefined.

Undefined

Since is of indeterminate form, apply L’Hospital’s Rule. L’Hospital’s Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.

Differentiate the numerator and denominator.

Differentiate using the Power Rule which states that is where .

Use to rewrite as .

Differentiate using the chain rule, which states that is where and .

To apply the Chain Rule, set as .

Differentiate using the Power Rule which states that is where .

Replace all occurrences of with .

To write as a fraction with a common denominator, multiply by .

Combine and .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply by .

Subtract from .

Move the negative in front of the fraction.

Combine and .

Move to the denominator using the negative exponent rule .

By the Sum Rule, the derivative of with respect to is .

Differentiate using the Power Rule which states that is where .

Since is constant with respect to , the derivative of with respect to is .

Add and .

Combine and .

Combine and .

Cancel the common factor.

Rewrite the expression.

Multiply the numerator by the reciprocal of the denominator.

Rewrite as .

Multiply by .

Divide the numerator and denominator by the highest power of in the denominator, which is .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Split the limit using the Limits Quotient Rule on the limit as approaches .

Move the limit under the radical sign.

Split the limit using the Sum of Limits Rule on the limit as approaches .

Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .

Evaluate the limit of which is constant as approaches .

Evaluate the limit of which is constant as approaches .

Divide by .

Add and .

Any root of is .

Evaluate Using L’Hospital’s Rule limit as x approaches infinity of x/( square root of x^2+1)