Question

Writing Use a graphing utility to graph \(f(x)=x, \quad g(x)=\sin x, \quad\) and \(\quad h(x)=\frac{\sin x}{x}\) in the same viewing window. Compare the magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is "close to" \(0 .\) Use the comparison to write \(a\) short paragraph explaining why \(\lim _{x \rightarrow 0} h(x)=1\).

Short answer

For values of \(x\) close to 0, \(f(x) = x\) approaches 0 and \(g(x) = \sin x\) approaches \(x\). Thus function \(h(x)=\frac{\sin x}{x}\) will be approximately equal to 1. Hence, \(\lim_{x \rightarrow 0} h(x)=1\).

Step-by-step solution

Graph the functions

Graph \(f(x)=x\), \(g(x)=\sin x\), and \(h(x)=\frac{\sin x}{x}\) using a graphing utility tool. By doing this, you'll able to see behavior of functions visually.

Compare magnitudes of \(f(x)\) and \(g(x)\) when \(x\) is close to 0.

Indicate the point '0' on the x-axis. Observe and compare the values of \(f(x)\) and \(g(x)\) when \(x\) is near 0. By doing this, you notice that \(f(x) = x\) approaches 0 and \(g(x) = sinx\) approaches \(x\) when \(x\) is close to 0.

Use comparison to explain \(\lim_{x \rightarrow 0} h(x)=1\)

In the case of the function \(h(x)=\frac{\sin x}{x}\), as \(x\) gets close to 0, the value of sine \(x\) and \(x\) are close to equal. Based on this observation, \(\frac{\sin x}{x}\) approximately equals to 1 for values of \(x\) near 0. Thus, we can say that \(\lim_{x \rightarrow 0} h(x)=1\).

Key concepts

Graphing Utility

Graphing utilities are powerful tools that allow us to visualize and understand the behavior of functions. They take complex mathematical expressions and transform them into visual graphs that we can easily interpret. This is incredibly useful in calculus, particularly when dealing with functions whose behaviors are not immediately obvious from their equations alone.

For instance, when graphing the functions in our exercise—namely, f(x) = x, g(x) = sin x, and h(x) = sin x / x—a graphing utility can help us compare their magnitudes visually. This comparison is especially important near critical points, such as x = 0, where traditional algebraic methods might be less effective.

It's important when using a graphing utility to choose an appropriate scale and viewing window to accurately capture the behavior of the functions around the points of interest.

Sine Function

The sine function, represented as sin x, is one of the fundamental trigonometric functions, and it plays a crucial role in various areas of mathematics, including calculus. It is periodic and oscillates between -1 and 1. The sine function describes a smooth, continuous wave that repeats every 2π units.

Understanding the sine function's behavior is particularly important when evaluating limits that involve trigonometric expressions. In our exercise, we see that as x approaches zero, the sine function's output also approaches the value of x. This is a special property of the sine function around zero that can be visualized through graphing and is instrumental to understanding the limit of the function h(x).

Limit of a Function

In calculus, the limit of a function is a fundamental concept that describes the behavior of that function as it approaches a certain point. The notation \lim_{x \rightarrow c} f(x) denotes the value that f(x) approaches as x gets close to c.

Limits help us make sense of values that might not be directly computable, such as when a function is undefined at a certain point. For our function h(x) = sin x / x, as x approaches zero, neither sin x nor x can be evaluated at zero. However, by understanding their behaviors near zero, we can conclude that the limit of h(x) as x approaches zero is 1. This is essential for understanding the continuity and differentiability of functions in calculus.

Comparing Magnitudes

Comparing magnitudes in calculus involves examining the relative sizes of function values as the input approaches a particular point. This comparison can often shed light on the behavior of more complex functions. When the magnitudes of two functions become increasingly close as the input nears a certain value, it can indicate that one function is approaching another in terms of its rate of growth or decrements.

In our exercise, comparing the magnitudes of f(x) = x and g(x) = sin x as x moves close to zero shows that they become nearly identical. This observation is pivotal to concluding why h(x) = sin x / x has a limit of 1 at zero. It indicates a fundamental concept in calculus, which is that the behavior of functions near a point can determine their limits, despite what might be happening at the point itself.