Question

\(\lim _{x \rightarrow 0} \frac{1-\cos ^{3} x}{x \sin x \cdot \cos x}\) is equal to (1) \(\frac{2}{3}\) (2) \(\frac{3}{5}\) (3) \(\frac{3}{2}\) (4) \(\frac{3}{4}\)

Short answer

\(\frac{3}{2}\)

Step-by-step solution

Identify the limit expression

We need to find the limit \ \ \ \( \lim _{x \rightarrow 0} \frac{1-\cos ^{3} x}{x \sin x \cdot \cos x} \)

Simplify the numerator using trigonometric identities

Recall that \ \ \ \( 1 - \cos^3 x \) can be written using the identity for the difference of cubes. \ \ \ \[ 1 - \cos^3 x = (1 - \cos x)(1 + \cos x + \cos^2 x) \] \ \ Substituting this into the limit expression gives: \ \ \ \( \lim _{x \rightarrow 0} \frac{(1 - \cos x)(1 + \cos x + \cos^2 x)}{x \sin x \cdot \cos x} \)

Divide and Simplify

Notice that \( \frac{1 - \cos x}{x} \) is a common limit form. As \( x \rightarrow 0 \), \( \frac{1 - \cos x}{x^2} \rightarrow \frac{1}{2} \). So re-write the limit as: \ \ \ \( \lim _{x \rightarrow 0} \frac{(1 - \cos x) \cdot (1 + \cos x + \cos^2 x)}{x \cdot x \sin x \cdot \cos x} \rightarrow \lim _{x \rightarrow 0} \frac{(1 - \cos x)}{x^2} \cdot \frac{(1 + \cos x + \cos^2 x)}{\sin x \cdot \cos x} \)

Evaluate the limit

Evaluate each part of the fraction independently. Use \( \frac{1 - \cos x}{x^2} \rightarrow \frac{1}{2} \). For small \( x \), \( \sin x \sim x \) and \( \cos x \sim 1 \), so \( \frac{\sin x \cdot \cos x}{x \cdot 1} \rightarrow 1 \). Thus, the limit is: \ \ \ \( \lim _{x \rightarrow 0} \frac{(1 - \cos x)}{x^2} \cdot \frac{(1 + \cos x + \cos^2 x)}{\sin x \cdot 1} = \frac{1}{2} \times (3) = \frac{3}{2} \).

Key concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus and algebra. They allow us to simplify and transform expressions involving trigonometric functions like sine, cosine, and tangent.
In this problem, we use the identity for the difference of cubes to simplify the numerator. Let's break it down:
The identity for the difference of cubes is given by:

• .

This transformation is crucial as it makes the expression easier to work with when evaluating the limit.
Understanding and effectively using trigonometric identities can greatly help in solving complex calculus problems.
Remember, these identities are powerful tools and can simplify seemingly complicated calculations.
Take your time to familiarize them as they will frequently appear in various mathematical contexts.

Limit Evaluation

Limit evaluation is a central concept in calculus. It involves finding the value that a function approaches as the input approaches a certain point.
In this problem, we aim to evaluate the limit as x approaches 0 for the given function:

• 
  The main challenge here is to simplify the expression to a form where the limit can be evaluated.
  By breaking down the expression and using known limits, we can solve the problem step by step.
  For instance, using the known limit: allows us to approach the problem methodically.
  Understanding these fundamental limits and their properties can help solve many calculus problems efficiently.

Difference of Cubes

The difference of cubes identity is a special algebraic tool used to factor expressions of the form Given by the formula: In this exercise, we used this identity to simplify the numerator:

• Breaking down complex expressions into simpler parts often makes it easier to evaluate limits.
  It's important to recognize opportunities to apply such identities as they often lead to significant simplifications.
  Practicing these algebraic manipulations can be very helpful in dealing with advanced calculus problems.