Problem 33 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

\(\lim _{x \rightarrow 0} \frac{\log x^{n}-[x]}{[x]}, n \in N,([x]\) denotes greatest integer less than or equal to \(x\) ) [2002] (a) has value \(-1\) (b) has value 0 (c) has value 1 (d) does not exist

Step-by-Step Solution

Answer

(d) does not exist

Step 1: Understand the Function Components

The function is given as \( \frac{\log x^n - [x]}{[x]} \), where \([x]\) is the greatest integer less than or equal to \(x\). This means \([x]\) gives the integer part of \(x\). For \(0 < x < 1\), \([x] = 0\). For larger values, it returns floor value.

Step 2: Simplify the Expression as \(x \to 0\)

For \(0 < x < 1\), \([x] = 0\), so our expression simplifies to \(\frac{\log x^n}{0}\), which is undefined. We should consider values of \(x\) approaching 0 but greater than 1. In this case, \([x] = k\) where \(k\) is an integer.

Step 3: Consider the Case for \(x > 1\)

In this region (\(x>1\)), \([x] = k\), such that \(x \approx k\). We rewrite the expression as \(\frac{\log(k^n) - k}{k} = \frac{n\log k - k}{k} = n \frac{\log k}{k} - 1\). As \(x \to 0\), this limit is not defined for \(x>1\).

Step 4: Evaluate Limit \(x \to 0\) for Small \(x\)

Let \(x \to 0^+\) implies \(x\) just slightly greater than 0, here \([x] = 0\), making the term \(\frac{\log(x^n)}{[x]}\) undefined due to division by zero. Hence, the limit does not exist.

Key Concepts

Greatest Integer Function

The greatest integer function, often represented as \([x]\), plays a crucial role in many mathematical contexts. It is essential to understand what this function does to effectively handle problems involving it. \([x]\) denotes the largest integer less than or equal to \(x\). This means that it rounds down \(x\) to the nearest whole number.
For example:

• If \(x=2.7\), then \([x]=2\).
• If \(x=-1.3\), then \([x]=-2\), since it rounds down in the negative direction as well.

To intuit how \([x]\) behaves, imagine stepping down on a stair where each step represents an integer. This function can sometimes also be referred to as the "floor" function. In problems involving limits, \([x]\) often causes abrupt changes in values as \(x\) crosses integer boundaries, making it crucial to examine closely, as shown in this exercise.

Logarithmic Functions

Logarithmic functions, expressed typically as \(log x\), are a significant concept in calculus and beyond. They represent the power to which a base (often 10 or \(e\), the natural logarithm base) must be raised to obtain a given number. For any positive number \(a\), \(log_a x \,= \(y\)\) means that \(a^y = x\).
Some key properties include:

• \(log(1) = 0\), because any base raised to the power of zero is 1.
• \(log(ab) = log a + log b\), highlighting its additive property.
• \(log(frac{a}{b}) = log a - log b\).

When solving for limits involving logarithmic expressions like \(\lim{x \to 0} \log x^n\), notice how the behavior of the logarithm dramatically changes as \(x\) approaches sensitive values like 0 or 1. The logarithm of values close to zero tends to \(\infty -\), indicating rapid decline. This can cause expressions to become undefined, as seen in the division by zero in this limit problem.

Undefined Expressions

In calculus, determining when expressions become undefined is pivotal, especially when examining limits. An expression is undefined when basic arithmetic rules do not give a precise value. Mainly, this happens when dividing by zero. In the given problem, the expression \(\frac{\log x^n - [x]}{[x]}\) becomes undefined as \(x o 0\) since \([x]=0\).
When dividing by zero, by definition, we cannot assign a finite number, since no quantity can multiply zero to yield any non-zero result.
Thus, such cases often lead to indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) which require further analysis:

• Tools like L'Hopital's rule can sometimes reinterpret these forms through derivatives, provided certain conditions apply.
• Rewriting expressions, factoring, or using limits laws often illuminate these undefined zones more clearly.

Knowing when expressions don't yield a definite outcome is vital, especially in limits, as it guides us to conclude whether a limit exists or if it is, indeed, undefined, as demonstrated in this exercise.