Problem 5 - Calculus of a Single Variable

Question

Estimating a Limit Numerically In Exercises 1–6, complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 0} \frac{\sin x}{x} $$

Step-by-Step Solution

Answer

The limit of \(\lim _{x \rightarrow 0} \frac{\sin x}{x} = 1\)

Step 1: Apply a known limit

The limit \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\) is a standard limit in calculus. It is known to equal 1. This is true because as x gets infinitely close to zero, \(\sin x\) also approaches x. Therefore, the fraction \(\frac{\sin x}{x}\) approaches 1.

Step 2: Using L'Hopital's rule

One way to show why this limit equals to 1 is by applying L'Hopital's rule. This rule states that if the limit of a function results in a 0/0 or ±∞/±∞ form, the limit can be computed as the limit of the ratios of their derivatives. Applying L'Hopital's rule to \(\frac{\sin x}{x}\) gives \(\frac{\cos x}{1}\), since the derivative of sin(x) is cos(x) and the derivative of x is 1. Thus, as x approaches 0, cos(x) approaches 1, so the limit is 1.

Step 3: Graph the function

Graphing the function \(\frac{\sin x}{x}\) confirms that as x approaches 0 the function approaches 1. Near 0, the graph of the function gets closer and closer to 1, demonstrating graphically why the limit as x approaches 0 of \(\frac{\sin x}{x}\) is 1. However, note that the function is not defined at x=0.

Key Concepts

L'Hopital's Rule

L'Hopital's Rule is a valuable tool in calculus for finding limits of indeterminate forms. Sometimes when you substitute a value into a limit, you get an uncertain form like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\). When this happens, L'Hopital's Rule can help. Simply put:

• Check if the limit gives an indeterminate form.
• Take the derivative of the numerator and the derivative of the denominator separately.
• Recompute the limit using these derivatives.

For example, with \(\lim _{x \rightarrow 0} \frac{\sin x}{x}\), substituting \(x = 0\) gives \(\frac{0}{0}\), an indeterminate form. By taking derivatives, the problem changes into \(\lim _{x \rightarrow 0} \frac{\cos x}{1}\). Since \(\cos x\) equals 1 when \(x\) is 0, the limit simplifies easily to 1. Thus, L'Hopital's Rule elegantly confirms the limit found from this calculus problem as 1.

Trigonometric Limits

Trigonometric limits often occur in calculus when evaluating expressions involving sine, cosine, and other trigonometric functions as the variable, usually \(x\), approaches a particular value.One of the most fundamental of these is \(\lim _{x \rightarrow 0} \frac{\sin x}{x} \), which equals 1. This is based on the properties of the sine function and its behavior around zero.Here’s why it’s important:

• The sine function, for very small values of \(x\), behaves very closely to the line \(y = x\).
• Therefore, the division \(\frac{\sin x}{x}\) gets closer and closer to 1, as \(x\) nears zero.
• This kind of limit helps in approximations and solutions in calculus and forms the basis for more complex evaluations.

Understanding this limit can make other trigonometric limits more intuitive and easier to handle.

Graphing Functions

Graphing functions is a visual way to understand how functions behave. By plotting points of a function, you can see how it behaves as the input, such as \(x\), changes.When graphing \(\frac{\sin x}{x}\), you will notice a curve that approaches but never quite touches 1 as \(x\) nears zero. This is significant:

• The graph shows how near-zero values of \(x\) lead to values of the function getting closer to 1.
• It's a great way to confirm numerical results from calculus.
• Understanding the graph helps in visualizing why the function behaves the way it does near a specific point.

However, remember that although the graph shows the tendency of the function, \(\frac{\sin x}{x}\) is undefined exactly at \(x = 0\). But still, you can visually interpret the limit, as the expression trends towards 1.