Problem 63 - Single Variable Calculus

Question

Evaluate \(\lim _{x \rightarrow 0} \frac{\sin x-x+\frac{1}{6} x^{3}}{x^{5}}\). Hint: Use the Maclaurin series representation of \(\sin x\).

Step-by-Step Solution

Answer

The Maclaurin series for sin(x) is given by \(\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\). Substituting the Maclaurin series of sin(x) into the given expression and simplifying, we get \(\frac{\frac{x^5}{5!}}{x^5}\). Evaluating the limit as x approaches 0, we find that the limit is \(\frac{1}{120}\).

Step 1: Write down the Maclaurin series for sin(x)

The Maclaurin series for sin(x) is given by: \[\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots\]

Step 2: Substitute the Maclaurin series of sin(x) into the expression

Now, we will substitute the Maclaurin series of sin(x) into the given expression and simplify: \[\frac{\sin x-x+\frac{1}{6} x^{3}}{x^{5}} =\frac{(x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots)-x+\frac{1}{6} x^{3}} {x^5}\]

Step 3: Simplify the expression

Next, we will simplify the expression by combining like terms and canceling out common terms: \[\frac{-\frac{1}{6}x^3+\frac{1}{6}x^{3}+\frac{x^5}{5!}}{x^5}=\frac{\frac{x^5}{5!}}{x^5}\]

Step 4: Evaluate the limit

Finally, we will evaluate the limit as x approaches 0: \[\lim_{x\to0} \frac{\frac{x^5}{5!}}{x^5} = \lim_{x\to0} \frac{1}{5!}=\frac{1}{120}\] Thus, the limit of the given function as x approaches 0 is \(\frac{1}{120}\).

Key Concepts

Limits

In calculus, limits help us understand the behavior of functions at specific points, often where direct computation is not possible. The expression \ \(\lim_{x \rightarrow 0}\frac{\sin x-x+\frac{1}{6} x^{3}}{x^{5}} \ \) asks us to find the value that the given function approaches as \(x\) gets very close to 0. Limits are foundational in calculus, providing a way to define derivatives and integrals. They are crucial for understanding continuous lines and curves and handle situations that involve very small or very large values. Here, by inserting and simplifying the Maclaurin series, we can find the behavior of the function near \(x = 0\).

Trigonometric Functions

Trigonometric functions like \(\sin(x)\) are important in mathematics, especially within calculus. They describe relationships between the angles and sides of triangles and appear repeated many places including oscillations and waves. The function \(\sin(x)\) can also be expanded using the Maclaurin series, which represents the function as an infinite sum of power terms. In our problem, the Maclaurin series expansion provides an accurate polynomial approximation for \(\sin(x)\) at values close to zero. This technique simplifies complex expressions, allowing us to make precise calculations in calculus problems involving limits and derivatives.

Single Variable Calculus

Single variable calculus deals with functions of one variable and includes concepts like limits, derivatives, and integrals. The exercise of evaluating a limit using a Maclaurin series falls squarely within this domain. By understanding the behavior of functions close to a point, calculus allows us to handle changes and areas under curves.

• Limits: Understand functions at specific points.
• Derivatives: Determine rates of change.
• Integrals: Measure areas under curves.

By mastering these concepts, students analyze compound relationships and solve practical and dynamic problems with ease.

Series Expansion

Series expansions, like the Maclaurin series, transform complex functions into simpler polynomial forms, making difficult operations manageable. These expansions approximate functions over specific intervals and are useful in calculus and analysis. By expanding \(\sin(x)\) into its Maclaurin series, we simplify our problem to evaluate the limit near zero. When using series expansions, it's important to understand

• Convergence: Does the series approach a specific value?
• Accuracy: How close is the approximation to the actual function?

These elements ensure that series expansions reliably represent functions, allowing for precise calculations within the scope of calculus.