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

Evaluate the limit of the numerator and the limit of the denominator.
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.
Find the derivative of the numerator and denominator.
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 .
Combine and .
Combine and .
Cancel the common factor.
Rewrite the expression.
Multiply the numerator by the reciprocal of the denominator.
Rewrite as .
Multiply by .
Take the limit of each term.
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 limits by plugging in the value for the variable.
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .