Calculus Examples

Evaluate the Limit limit as x approaches infinity of ( natural log of x)/x
limxln(x)x
Step 1
Apply L'Hospital's rule.
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Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
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Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
limxln(x)limxx
Step 1.1.2
As log approaches infinity, the value goes to .
limxx
Step 1.1.3
The limit at infinity of a polynomial whose leading coefficient is positive is infinity.
Step 1.1.4
Infinity divided by infinity is undefined.
Undefined
Step 1.2
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.
limxln(x)x=limxddx[ln(x)]ddx[x]
Step 1.3
Find the derivative of the numerator and denominator.
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Step 1.3.1
Differentiate the numerator and denominator.
limxddx[ln(x)]ddx[x]
Step 1.3.2
The derivative of ln(x) with respect to x is 1x.
limx1xddx[x]
Step 1.3.3
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
limx1x1
limx1x1
Step 1.4
Multiply the numerator by the reciprocal of the denominator.
limx1x1
Step 1.5
Multiply 1x by 1.
limx1x
limx1x
Step 2
Since its numerator approaches a real number while its denominator is unbounded, the fraction 1x approaches 0.
0
 [x2  12  π  xdx ]