Question

Approximation (a) Find \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}\) (b) Use your answer to part (a) to derive the approximation \(\cos x \approx 1-\frac{1}{2} x^{2}\) for \(x\) near 0 . (c) Use your answer to part (b) to approximate \(\cos (0.1)\) . (d) Use a calculator to approximate \(\cos (0.1)\) to four decimal places. Compare the result with part (c).

Short answer

\(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}} = \frac{1}{2}\), \(\cos x \approx 1 - \frac{1}{2} x^{2}\) for x near 0, \(\cos(0.1) \approx 0.995\), and using the calculator, \(\cos(0.1) = 0.9950\).

Step-by-step solution

Apply L'Hopital's Rule

Apply L'Hopital's rule which states that \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\) when the limit yields an indeterminate form. So, we have \(\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}} = \lim _{x \rightarrow 0} \frac{\sin x}{2x}\). Applying L'Hopital's rule again gives \(\lim _{x \rightarrow 0} \frac{\cos x}{2} = \frac{1}{2}\).

Derive the Approximation for cos(x)

Using the limit derived in step 1, rewrite the cos(x) equation as \(\cos x = 1 - \frac{1}{2} x^{2}\) for x near 0.

Approximate the Value of cos(0.1)

Substitute 0.1 in place of x in the equation from step 2 to approximate the value of cos(0.1). So, we find that \(\cos(0.1) \approx 1 - \frac{1}{2} * 0.1^{2} = 0.995\)

Calculate the Value of cos(0.1) to Compare

Use a calculator to find \(\cos(0.1)\) to four decimal places. The result is 0.9950.

Key concepts

Understanding L'Hopital's Rule

When faced with complex calculus problems, L'Hopital's Rule is a powerful tool for evaluating limits that result in indeterminate forms such as 0/0 or \[ \infty/\infty \]. It essentially tells us that if we have a limit of the form \(\lim_{x \to a} \frac{f(x)}{g(x)}\) that gives an indeterminate form, we can instead look at the limit of the derivatives of the numerator and the denominator. Formally, it states \(\lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}\), assuming certain conditions are met and the right-hand limit exists.

In our exercise, applying L'Hopital's Rule facilitated the simplification of the given limit by turning the indeterminate form into a simpler quotient of derivatives, which could then be easily evaluated. It's a rule that often comes in handy when dealing with tricky calculus limits and helps students bypass more complex algebraic manipulation or intuitive approaches like graphing.

Cosine Approximation Near Zero

Cosine approximation is an essential concept when trying to estimate the value of the cosine function for values close to zero without using a calculator. It relies on the Taylor Series expansion for the cosine function, which is an infinite sum. However, near zero, the function can be approximated by the first few terms of this series. Specifically, the approximation \(\cos x \approx 1 - \frac{1}{2} x^{2}\) is derived by considering only the first two non-zero terms of the Taylor expansion of the cosine function.

This approximation is particularly useful because it provides a straightforward and computationally inexpensive way to estimate \(\cos x\) for small values of \(x\). In our exercise, using this approach significantly simplified the process of estimating \(\cos(0.1)\) and helped in understanding how close the approximation is to the actual value.

The Limit of a Function

The concept of the limit of a function is foundational in calculus and represents the value that a function approaches as the input approaches some point. Limits allow us to understand the behavior of functions at points where they might not be well-defined or at points approaching infinity. In our original exercise, finding the limit as \(x\) approaches zero helps to establish the behavior of \(\frac{1-\cos x}{x^{2}}\) when \(x\) is infinitesimally close to zero but not exactly at zero.

The powerful aspect of limits is their ability to generalize a function's behavior around a point, which can be instrumental in many areas of analysis, including continuity, differentiability, and integral calculus. Identifying and applying the proper techniques to evaluate limits is an essential skill for students tackling problems in calculus.

Dealing with Indeterminate Forms

Indeterminate forms often occur in calculus when evaluating limits and can be deceptive as they don't readily provide an answer. Forms such as 0/0, \(\infty/\infty\), \(\infty - \infty\), \(0 \times \infty\), \(\infty^{0}\), \(0^{0}\), and \(1^{\infty}\) are all considered indeterminate. In the case of our exercise, the original form \(\frac{1-\cos x}{x^{2}}\) as \(x\) approaches zero yields 0/0, which is indeterminate. Without additional techniques, we cannot decide what the limit is from the form alone.

Methods like L'Hopital's Rule, algebraic simplification, or polynomial expansion are often employed to resolve these forms into something more manageable. Understanding how to recognize and address indeterminate forms enables students to navigate through complex limit problems efficiently and arrive at precise solutions.