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Chapter 1: Problem 1

Find the discontinuities, if any. $$ f(x)=\sin \left(x^{2}-2\right) $$

Short Answer

Expert verified
The function is continuous everywhere; there are no discontinuities.

Step by step solution

01

Understanding the Function

First, recognize that the function given is a sine function: \(f(x) = \sin(x^2 - 2)\). Sine functions, by nature, are continuous everywhere because their domain is all real numbers.
02

Analyzing the Argument of the Sine Function

Examine the argument of the sine function: \(x^2 - 2\). This is a quadratic function, and it is also continuous for all real numbers because polynomials are continuous everywhere.
03

Concluding the Continuity of the Function

Since both the sine function and its argument \(x^2 - 2\) are continuous for all real numbers, the composition of these two functions, \(f(x) = \sin(x^2 - 2)\), is continuous for all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Continuous Functions
A function is continuous if, loosely speaking, you can trace the graph of the function without lifting your pencil from the paper. In mathematical terms, a function is continuous at a point if the limit of the function as it approaches the point from both the left and the right equals the function's value at that point.

Continuous functions are important because they behave predictably鈥攖hey do not have abrupt jumps or breaks. This predictability is crucial in many areas of mathematics and its applications, such as calculus and physics. When dealing with continuous functions, calculating limits, integrals, and derivatives becomes much more straightforward.
  • They have no sudden jumps.
  • Their graphs can be drawn without lifting the pen.
  • Main property: \( ext{lim}_{x \to c} f(x) = f(c)\) for all points \(c\) in their domain.
Sine Function
The sine function is a trigonometric function that is foundational to understanding wave patterns and circles. It is defined for all real numbers and is periodic with a period of \(2\pi\). This means that it repeats its values in intervals of \(2\pi\) along the x-axis.

One of the most remarkable behaviors of the sine function is that it is continuous for every real number. This continuity is due to the smooth and wave-like behavior of sine, which ensures there are no breaks or gaps in its graph.
  • Domain: all real numbers.
  • Range: \([-1, 1]\).
  • Period: \(2\pi\).
  • Smooth and continuous everywhere.
Polynomial Continuity
Polynomials form a basis for many functions and are some of the most straightforward functions in mathematics. They are represented by algebraic expressions like \(x^2 - 2\), and are continuous for all real numbers. This makes them very predictable and useful in mathematical analysis.

The continuity of polynomials means that their graphs have no breaks, jumps, or holes. This property is very handy when you need to graph them, as their curves smoothly transition between points. Polynomials of all degrees are continuous, from constant terms up through linear, quadratic, cubic, and higher.
  • Continuous on the entire real line.
  • Expressed as a sum of terms \(a_nx^n + a_{n-1}x^{n-1} + ... + a_0\).
  • Each term in a polynomial is continuous.
  • Results in overall continuity of the polynomial function.

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Most popular questions from this chapter

Find the limits. $$ \lim _{h \rightarrow 0} \frac{\sin h}{1-\cos h} $$

Let \(p(x)\) and \(q(x)\) be polynomials, with \(q\left(x_{0}\right)=0\). Discuss the behavior of the graph of \(y=p(x) / q(x)\) in the vicinity of \(x=x_{0}\). Give examples to support your conclusions.

In each part, find the largest open interval centered at \(x=1\), such that for each \(x\) in the interval, other than the center, the value of \(f(x)=1 /(x-1)^{2}\) is greater than \(M\). (a) \(M=10\) (b) \(M=1000\) (c) \(M=100,000\)

Find the limits. $$ \lim _{\theta \rightarrow 0} \frac{\sin ^{2} \theta}{\theta} $$

First rationalize the numerator and then find the limit. $$ \lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{x} $$
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