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How do you find the limit of \[\dfrac{1}{\ln x}\] as x approaches infinity?

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Last updated date: 20th Jun 2024
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Answer
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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 polynomial. Sometimes we use some kind of radical conjugate. The value of the inverse of infinity is zero. If we have to find the limit something in the form (constant)/(some expression) then we can simply find the limit of the denominator. Also, the \[\ln x\] graph tends to infinity when x approaches infinity.

Complete step by step answer:
In the given question, we have to find the limit of \[\dfrac{1}{\ln x}\] when x approaches to zero.
That is, \[\Rightarrow \displaystyle \lim_{x\to \infty }\dfrac{1}{\ln x}\]. The limiting condition of x is infinity which is basically an undefined number and also a large quantity.
Here the numerator is a constant term. So, we can directly find the limit of the denominator when x approaches infinity.
\[\Rightarrow \dfrac{1}{\displaystyle \lim_{x\to \infty }\ln x}\]
As we know that the graph of ln x tends to infinity when x tends to infinity. Then we can write
\[\Rightarrow \dfrac{1}{\infty }\]
We know that infinity is a large quantity which when divides 1 is approximately equal to 0.
Hence, the limit of \[\dfrac{1}{\ln x}\] tends to 0 as x approaches infinity.

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 \[\dfrac{1}{\ln x}\] when x approaches to infinity by the expansion method. In the expansion method, we expand \[\ln x\] as an algebraic expression in terms of powers of x. In any method we get the same answer but the simplicity of the question differs.