Problem 82 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

\(\lim _{x \rightarrow 0} x\left[\frac{1}{x}\right]\)

Step-by-Step Solution

Answer

The limit as x approaches 0 of \(x\left[\frac{1}{x}\right]\) is 1.

Step 1 Title - Look at the initial expression

Initially, our expression is \(x\left[\frac{1}{x}\right]\). But there is an indeterminate form when x approaches 0.

Step 2 Title - Simplify the expression

Notice that the x in the numerator and the x in the denominator can cancel each other out. This leaves us with the simplified expression of 1.

Step 3 Title - Substitute the value for the limit

Finally, we substitute x = 0 into the simplified expression. There are no more x's left, so we are simply left with 1.

Key Concepts

Indeterminate Forms

When we deal with limits in calculus, especially as a variable approaches a certain point, we often encounter what is known as indeterminate forms. These are forms where the limit cannot be directly determined from the initial expression and thus requires further manipulation.

Common indeterminate forms include \(0/0\), \(\infty/\infty\), \(0\cdot\infty\), \(\infty-\infty\), \(0^0\), \(\infty^0\), and \(1^\infty\). For example, in the exercise \(\lim _{x \rightarrow 0} x\left[\frac{1}{x}\right]\), we face the \(0/0\) indeterminate form, because as \(x\) approaches zero, both the numerator and denominator approach zero as well.

To resolve such forms, we often employ techniques like simplifying expressions, factoring, expanding, or L'Hôpital's Rule. The goal is to convert the indeterminate form into a determinate one, which can be evaluated more straightforwardly.

Simplifying Expressions

Simplifying expressions is a crucial step in finding limits that involve indeterminate forms. It involves reducing an expression to its simplest form, making it easier to understand and solve.

In our example \(\lim _{x \rightarrow 0} x\left[\frac{1}{x}\right]\), simplification is achieved by cancelling the \(x\) in the numerator with \(x\) in the denominator. This process leads to a much clearer expression: 1. To simplify, we look for common factors, use algebraic identities, or restructure the expression to eliminate complex parts.

Simplification can reveal the behavior of the function as it approaches the limit and often removes any complication that may arise from indeterminate forms.

Substitution in Limits

Once an indeterminate form has been simplified, we can often use substitution to evaluate the limit. This technique involves replacing the variable with the value it approaches in the limit.

In our exercise, after simplifying the expression, we arrive at a constant, 1. Since we've removed the variable \(x\), we can directly substitute \(x = 0\) without affecting the expression's value.

Substitution is particularly helpful when the limit leads to a constant or when the function becomes continuous post-simplification. However, if substitution after simplification still leads to an indeterminate form, further techniques may need to be employed to evaluate the limit.