Calculus Examples

Evaluate Using L'Hospital's Rule limit as x approaches infinity of ( natural log of x)/x
Step 1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1
Take the limit of the numerator and the limit of the denominator.
Step 1.2
As log approaches infinity, the value goes to .
Step 1.3
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Step 1.4
Infinity divided by infinity is undefined.
Undefined
Step 2
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.
Step 3
Find the derivative of the numerator and denominator.
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Step 3.1
Differentiate the numerator and denominator.
Step 3.2
The derivative of with respect to is .
Step 3.3
Differentiate using the Power Rule which states that is where .
Step 4
Multiply the numerator by the reciprocal of the denominator.
Step 5
Multiply by .
Step 6
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .