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Question: How do you find the limit \( \dfrac{{\ln x}}{x} \) where \( x \to \infty \) ?...

How do you find the limit ln⁡xx\dfrac{{\ln x}}{x} where x→∞x \to \infty ?

Explanation

Solution

Hint : Determining the limits algebraically can be in many ways. In case of polynomials, we can obtain the limits by simply substituting the limiting value of x into the polynomials. Sometimes we use some kind of radical conjugate. The value of the inverse of infinity is zero. If we have to find the limit of something where both the numerator and denominator approaches to infinity the the first thing that should come into someone's mind is to convert the equation into 00\dfrac{0}{0} form by substituting x=1xx = \dfrac{1}{x}

Complete step-by-step answer :
In the given question, we have to find the limit of ln⁡xx\dfrac{{\ln x}}{x} when x approaches infinity.
That is , ⇒lim⁡x→∞ln⁡xx\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{{\ln x}}{x} . The limiting condition of x is infinity which is basically an undefined number and also a large quantity.
Therefore we have a ratio of two infinities ∞∞\dfrac{\infty }{\infty } meaning that we will have to apply L’Hospital’s Rule.
⇒lim⁡x→∞ln⁡xx\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{{\ln x}}{x}
⇒lim⁡x→∞ddx(ln⁡x)ddx(x)\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{{\dfrac{d}{{dx}}\left( {\ln x} \right)}}{{\dfrac{d}{{dx}}\left( x \right)}}
⇒lim⁡x→∞1x1\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{{\dfrac{1}{x}}}{1}
⇒lim⁡x→∞1x\Rightarrow \mathop {\lim }\limits_{x \to \infty } \dfrac{1}{x} =0= 0
Therefore, when the limit approaches 0 because 1 divided over something approaching ∞\infty becomes closer and closer to zero.
For example, consider:
110=0.1\dfrac{1}{{10}} = 0.1
1100=0.01\dfrac{1}{{100}} = 0.01
11000=0.001\dfrac{1}{{1000}} = 0.001
We can see that as the denominator gets larger and larger, approaching ∞\infty , the value gets smaller and smaller the more closer to 0.
So, the correct answer is “0”.

Note : While solving such types of problems we have to concentrate on the left side and right side of the limiting value. We can also find the limit of ln⁡xx\dfrac{{\ln x}}{x} when x approaches infinity by the expansion method. In the expansion method, we can expand ln⁡x\ln x as an algebraic expression in terms of powers of x. In any method we will get the same answer but the simplicity of the question differs.