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Calculus Examples
limx→∞ln(x)x
Step 1
Step 1.1
Evaluate the limit of the numerator and the limit of the denominator.
Step 1.1.1
Take the limit of the numerator and the limit of the denominator.
limx→∞ln(x)limx→∞x
Step 1.1.2
As log approaches infinity, the value goes to ∞.
∞limx→∞x
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.
limx→∞ln(x)x=limx→∞ddx[ln(x)]ddx[x]
Step 1.3
Find the derivative of the numerator and denominator.
Step 1.3.1
Differentiate the numerator and denominator.
limx→∞ddx[ln(x)]ddx[x]
Step 1.3.2
The derivative of ln(x) with respect to x is 1x.
limx→∞1xddx[x]
Step 1.3.3
Differentiate using the Power Rule which states that ddx[xn] is nxn-1 where n=1.
limx→∞1x1
limx→∞1x1
Step 1.4
Multiply the numerator by the reciprocal of the denominator.
limx→∞1x⋅1
Step 1.5
Multiply 1x by 1.
limx→∞1x
limx→∞1x
Step 2
Since its numerator approaches a real number while its denominator is unbounded, the fraction 1x approaches 0.
0