Problem 463 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

$$ \text { Evaluate } \lim _{x \rightarrow 0} \frac{x-\sin x}{x^{3}} \text { without using series and } L \text { ' Hospital's rule. } $$

Step-by-Step Solution

Answer

The limit of the function as \(x\) approaches 0 is \(\frac{1}{6}\).

Step 1: Identify Indeterminate forms

First, identify the indeterminate form of the function. If you plug in \(x = 0\), you will get the indeterminate form \(0/0\), which means you can't directly substitute \(x\) with 0 in the function.

Step 2: Rewrite sin(x)

Rewrite the \(\sin(x)\) in the numerator. We know that \(\sin(x)\) can be written as \(x - x^{3}/3! + x^{5}/5! - x^{7}/7! + ...\). Hence, the equation can be rewritten as: \(\lim_{x \to 0} \frac{x - (x - x^{3}/3! + x^{5}/5! - x^{7}/7! + ...)}{x^{3}}\).

Step 3: Simplify the equation

If you simplify the equation, you will have \(\lim_{x \to 0} \frac{x^{3}/3! - x^{5}/5! + x^{7}/7! - ...}{x^{3}} = \lim_{x \to 0} \frac{1}{3!} - \frac{x^{2}}{5!} + \frac{x^{4}}{7!} - ...\).

Step 4: Evaluate the limit

Finally, if you substitute \(x = 0\) into the simplified equation, you will get \(\frac{1}{3!} = \frac{1}{6}\) as the answer.

Key Concepts

Indeterminate Forms

In calculus, when you evaluate the limit of a function as it approaches a particular point, you might encounter something known as an indeterminate form. One of the most common types of indeterminate forms is \(0/0\). This occurs when both the numerator and the denominator of a fraction approach zero as the limit is evaluated. You cannot simply substitute the value into the function in these cases because the mathematical operations become undefined.

• The key is to manipulate the function into a form where you can directly evaluate the limit.
• This is often done through algebraic manipulation, expansion, or simplification of the function.
• Identifying indeterminate forms is crucial as it guides you on how to proceed, such as whether additional steps, like factorization or applying specific limits techniques, are needed.

Understanding the nature of indeterminate forms helps you decide on the appropriate strategies to utilize for evaluating limits. This foundational step can make the difference in solving complex calculus problems successfully.

Trigonometric Limits

Trigonometric limits, involving functions like sine, cosine, and tangent, can present unique challenges due to their oscillating nature. Often, when dealing with trigonometric limits, especially as the variable approaches zero, you will find idiosyncrasies like those seen in the limit \( \lim _{x \to 0} \frac{\sin x}{x} = 1 \).
This specific limit plays a significant role in simplifying and evaluating various trigonometric limits.

• Understanding basic trigonometric limits is fundamental as they serve as building blocks for solving more complex problems.
• In the problem given, rewriting \(\sin(x)\) using its series expansion helps in simplifying the trigonometric expression to determine the limit.
• Trigonometric identities and expansions offer a powerful way to convert complex expressions into simpler forms.

This approach aligns perfectly with cases like \( \lim _{x \to 0} \frac{x - \sin x}{x^3} \), where arithmetic manipulation of series expansions leads to a solvable form.

Limit Evaluation Techniques

Evaluating limits in calculus often requires a strategic approach when direct substitution leads to an indeterminate form. Here are some common techniques that assist in evaluating these limits effectively:

• Series Expansion: This involves expanding a function in terms of its power series. For example, expanding \(\sin(x)\) provides a way to isolate terms to facilitate subsequent calculation steps like in the provided exercise.
• Algebraic Simplification: Simplifying the expression by combining like terms, factoring, or canceling out terms where possible can make it feasible to evaluate the limit.
• Advanced Techniques: While not used in this problem, methods like factoring, rationalizing, and conjugates might be necessary for other types of expressions.

Through these techniques, particularly series expansion in this case, you can transform a complex expression into an easily manageable one, allowing the calculation of the limit to be straightforward and intuitive.