Problem 49 - Essential Calculus

Question

In Exercises 49 and \(50,\) use the series representation of the function \(f\) to find \(\lim _{x \rightarrow 0} f(x)\) (if it exists). $$ f(x)=\frac{1-\cos x}{x} $$

Step-by-Step Solution

Answer

The value of \(\lim _{x \rightarrow 0} f(x)\) is 0.

Step 1: Express the function in terms of limits

The function is \(f(x) = \frac{1 - cos(x)}{x}\), which, as \(x \rightarrow 0\), takes the form \(0/0\). This indicates an indeterminate form, and hence, l'Hopital's rule can be applied. So, find the limit as \(x \rightarrow 0\) : \(\lim _{x \rightarrow 0} \frac{1 - cos(x)}{x}\)

Step 2: Apply l'Hopital's rule

According to l'Hopital's rule, the limit of a fraction where both the numerator and the denominator approach zero can be found by taking the derivative of the numerator and the denominator separately. So, differentiating \(1 - cos(x)\) with respect to \(x\) gives \(sin(x)\) and differentiating \(x\) with respect to \(x\) gives \(1\). Now, put these values back into the limit: \(\lim _{x \rightarrow 0} \frac{sin(x)}{1}\)

Step 3: Simplify the function and find the limit

With the new function \(\lim _{x \rightarrow 0} sin(x)\), which doesn't have an indeterminate form, you can directly substitute the value of \(x = 0\) into the function. \(sin(0) = 0\). Hence, \(\lim _{x \rightarrow 0} sin(x) = 0\)

Key Concepts

L'Hôpital's Rule

L'Hôpital's Rule is a very handy tool in calculus for tackling limits that result in an indeterminate form like \(0/0\) or \(\infty/\infty\). When you encounter these forms, applying L'Hôpital's Rule can simplify the process and help compute the limit. Here's how it works:

• You identify the indeterminate form. For example, \(f(x) = \frac{1 - \cos x}{x}\) as \(x \rightarrow 0\) becomes \(0/0\).
• Differentiation: Take the derivative of the numerator (\(1 - \cos(x)\)) and the derivative of the denominator (\(x\)). You get \(\sin(x)\) and \(1\), respectively.
• Replace the original function with the new fraction of derivatives: \(\lim _{x \rightarrow 0} \frac{\sin(x)}{1}\).
• Evaluate the new limit. Here, \(\lim _{x \rightarrow 0} \sin(x) = 0\), which wasn't an indeterminate form anymore.

It's crucial to use L'Hôpital's Rule properly. It requires both numerator and denominator to be differentiable and their limits to initially give indeterminate forms. This method significantly simplifies complex problems in calculus.

Trigonometric Limits

Trigonometric limits often serve as gateways to understanding more complex functions at a deeper level. They frequently occur in calculus, especially when dealing with derivatives and integrals of trigonometric functions.

• Trigonometric identities and limits are useful. For example, the limit \(\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1\) is a fundamental result in calculus and is commonly used to derive other trigonometric limits.
• Understanding trigonometric limits helps solve limits involving combinations like \(\frac{1 - \cos(x)}{x}\), which initially presents as \(0/0\). We use derivative techniques to tackle these limits effectively.
• Visual representations of these functions, such as graphs, can aid in understanding how trigonometric limits behave as they approach particular points.

Trigonometric limits serve as stepping stones for exploring various calculus phenomena and are essential for evaluating the behavior around certain critical points. They enhance the ability to solve complex analytical problems with ease.

Indeterminate Forms

Indeterminate forms arise when evaluating a limit and the result isn't immediately clear. The expression \(\frac{0}{0}\) is a classic example.

• Common indeterminate forms include \(0/0\), \(\infty/\infty\), \(0 \cdot \infty\), \(\infty - \infty\), \(1^\infty\), \(0^0\), and \(\infty^0\). Identifying these helps determine the technique needed to solve for limits.
• L'Hôpital's Rule can often resolve \(0/0\) and \(\infty/\infty\) forms by taking derivatives until a clear limit is reached. This was evident in the example \( \lim _{x \rightarrow 0} \frac{1 - \cos x}{x} \), where applying the rule helped simplify to \(\lim _{x \rightarrow 0} \sin(x)\).
• Comparative growth rates and algebraic manipulation are other techniques used to approach indeterminate forms. Rationalizing, factorization, and expansion are additional tools to tackle these scenarios.

Indeterminate forms often need careful analysis to resolve. Understanding these and the methods to handle them enhances problem-solving skills in calculus.