Problem 63 - Calculus: Graphical, Numerical, Algebraic

Question

Multiple Choice $$\lim _{x \rightarrow 0} \frac{\sin (3 x)}{x}=$$ (A) 1\(/ 3 \quad\) (B) 1 (C) 3 (D) \(\sin 3 \quad\) (E) does not exist

Step-by-Step Solution

Answer

The limit of \( \frac{\sin (3 x)}{x}\) as \(x \rightarrow 0\) is 3. So, the correct answer is (C) 3.

Step 1: Identify the indeterminate form

Firstly, check that at \(x \rightarrow 0\), the given expression is indeed in the form \(0/0\). Plugging \(x = 0\) into the equation \(\frac{sin(3x)}{x}\) gives \(\frac{sin(0)}{0}\) or \( \frac{0}{0}\).

Step 2: Apply L'Hopital's Rule

As we have the indeterminate form \( \frac{0}{0}\), we can apply L'Hopital's Rule, which says that the limit of a quotient of functions is equal to the limit of the derivatives of those functions. So, we will differentiate both the numerator and the denominator with respect to \(x\). The derivative of \(sin(3x)\) is \(3cos(3x)\), and the derivative of \(x\) is 1.

Step 3: Substitute \(x=0\) into the result

After applying L'Hopital's Rule, we get \(\frac{3cos(3x)}{1}\). Now, substitute \(x=0\) into the expression. The result is \(3cos(0)\), which simplifies to 3, as \(cos(0) = 1\).

Key Concepts

Limits in Calculus

Understanding limits is a foundational concept in calculus, which deals with the behavior of functions as they approach specific points or infinity. A limit attempts to find out what value a function approaches, as the input (or 'x' value) gets closer and closer to some number. For example, the expression \( \lim_{x \to 0} f(x) \) assesses the behavior of the function f(x) as x approaches 0.

Limits are essential for defining derivatives and integrals, which are core parts of differential and integral calculus. They are also crucial in determining continuity; a function is continuous at a point if the limit and the function value at that point are the same. When the function doesn't behave nicely (jumping around or approaching different values from different directions), the limit might not exist. However, even if a function seems 'misbehaved' (like going towards infinity), sometimes limits can still be determined using strategies such as factoring, conjugation, or the powerful L'Hopital's Rule in the case of indeterminate forms.

Sine Function Behavior

The sine function, denoted as \(\sin(x)\), is a periodic function that describes a smooth, wave-like pattern. It is fundamental in trigonometry and appears in many areas of mathematics and physics, especially in the study of waves and oscillations. The sine function takes an angle as an input and returns the ratio of the length of the side of the triangle opposite to the angle to the hypotenuse in a right-angled triangle.

The sine curve starts at 0, rises to a peak of 1, falls to -1, and returns to 0, completing what we call a 'cycle'. The behavior of \(\sin(x)\) around \(x=0\) is particularly interesting due to its 'linearity'. As x gets smaller, \(\sin(x)\) gets very close to the value of x itself. This property leads to the famous limit \(\lim_{x \to 0} \frac{\sin(x)}{x} = 1\), which is crucial for understanding the exercise and is intuitively understood by visualizing the unit circle and sine's behavior approaching the origin.

Indeterminate Forms

In calculus, we often encounter expressions that do not readily reveal their limit, referred to as 'indeterminate forms'. Common indeterminate forms include \(0/0\), \(\infty/\infty\), \(0\cdot\infty\), \(\infty - \infty\), \(0^0\), \(1^\infty\), and \(\infty^0\). These forms are not automatically undefined; rather, they need further analysis to determine the actual limit.

One of the most powerful tools for handling these indeterminate forms is L'Hopital's Rule. It says that if you have an indeterminate form of type \(0/0\) or \(\infty/\infty\), you can take the derivatives of the top and bottom functions and then take the limit again. The process can be repeated if necessary. In the context of our exercise, we used L'Hopital's Rule to transform the indeterminate form \(\frac{\sin(3x)}{x}\), as \(x \to 0\), into a form where the limit could be directly calculated. This concept is a testament to the ingenuity of mathematical problem-solving – when faced with a seemingly unsolvable expression, calculus gives us tools like L'Hopital's Rule to unlock the answer.