Problem 20 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(\lim _{x \rightarrow \infty} \frac{(\log x)^{2}}{x^{n}}, n>0\) is equal to (A) 1 (B) 0 (C) \(-1\) (D) \(\infty\)

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Expression

The given exercise is to find the limit of the expression \( \lim _{x \rightarrow \infty} \frac{(\log x)^{2}}{x^{n}} \), where \( n > 0 \).

Step 2: Analyze Growth Rates

Compare the growth rates of the numerator \((\log x)^{2}\) and the denominator \(x^n\). As \(x\) approaches infinity, the logarithmic function \((\log x)^{2}\) grows much slower than any positive power of \(x\).

Step 3: Apply Limit Properties

Since the polynomial function in the denominator \(x^n\) grows significantly faster than the slow logarithmic growth \((\log x)^2\), the expression \(\frac{(\log x)^{2}}{x^{n}}\) tends toward zero as \(x\) approaches infinity.

Step 4: Conclusion on Limit

Therefore, the limit \( \lim _{x \rightarrow \infty} \frac{(\log x)^{2}}{x^{n}} \) converges to 0.

Key Concepts

Growth Rates

Growth rates are an essential concept in understanding limits, particularly when comparing different types of functions. Growth rate refers to how rapidly a function increases or decreases as its variable approaches a certain point, often infinity. In the exercise, we examine the growth rates of two functions:

• Numerator: \((\log x)^{2}\)
• Denominator: \(x^n\)

Logarithmic functions like \((\log x)^2\) grow very slowly compared to polynomial functions like \(x^n\). As \(x\) becomes very large, polynomial terms typically increase much faster than logarithmic ones. This difference in growth rates is key to determining limits, as the faster-growing function in the denominator will push the ratio towards zero.
By analyzing growth rates, we discover that no matter how slowly the numerator grows, the denominator's rapid increase ensures that the ratio becomes tinier and tinier, confirming the limit approaches zero.

Logarithmic Functions

Logarithmic functions are integral in math as they represent the inverse operations of exponential functions. The logarithm \(\log x\) measures the number of times a base needs to be raised to produce \(x\). For practical use, logs help simplify large figures.One of the key characteristics of logarithmic functions is their slow growth. Even when squared, like \((\log x)^{2}\), they increase at a sluggish pace compared to other functions, such as polynomials. This characteristic was vital in the exercise, where the squared logarithm function appeared in the numerator.
When contrasted with polynomial functions, the slow-growing nature of logs plays a crucial role in solving limits. Even large values of the logarithmic functions remain overshadowed by the rapid pace of polynomials, guiding our conclusion that \(lim \frac{(\log x)^{2}}{x^{n}}\) approaches zero as \(x\) tends to infinity.

Polynomial Functions

Polynomial functions are expressions consisting of variables raised to different powers, often termed as degrees. The function \(x^n\) represents a polynomial of degree \(n\), where the value of \(n\) determines the complexity and growth of the polynomial.In limits and growth comparison, polynomial functions generally outpace many other types of expressions, including logarithmic functions. In the exercise, the denominator \(x^n\) signifies this polynomial form, which grows much more rapidly than the numerator's \((\log x)^2\).
This rapid growth makes polynomial functions dominant in limit contexts, especially when they are in the denominator. As in the exercise, the polynomial term in the denominator ensured the overall expression decreased toward zero, as its growth was unmatchable by the numerator's slow logarithmic increase.