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Show that the function $ f(x) = \left\{ \begin{array}{ll} x^4 \sin (1/x) & \mbox{if $ x \neq 0 $}\\ 0 & \mbox{if $ x = 0 $} \end{array} \right.$

is continuous on $ (-\infty, \infty) $.

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$f(x)=x^{4} \sin (1 / x)$ is continuous on $(-\infty, 0) \cup(0, \infty)$ since it is the product of a polynomial and a composite of atrigonometric function and a rational function. Now since $-1 \leq \sin (1 / x) \leq 1,$ we have $-x^{4} \leq x^{4} \sin (1 / x) \leq x^{4}$. Because $\lim _{x \rightarrow 0}\left(-x^{4}\right)=0$ and $\lim _{x \rightarrow 0} x^{4}=0,$ the Squeeze Theorem gives us $\lim _{x \rightarrow 0}\left(x^{4} \sin (1 / x)\right)=0,$ which equals $f(0) .$ Thus, $f$ is continuous at 0 and, hence, on $(-\infty, \infty)$.

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Oregon State University

Baylor University

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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This is problem number seventy one of this tour calculus eighth edition, Section two point five show that the function f of X equals except forthe time, sign of the quantity one over X X is not equal to zero and zero if X is equal to zero, is continuous on the domain of all real numbers and our we're going to use our definition of continuity to confirm our continuity. To make sure that we have confirmed continuity, Dad, for a function f and function F is only considered continuous if and only if the limit is X approaches. EVA function is equal to the function evaluated at eight. So if we take a look at the function, it is definitely continuous on all rials as is on this first function is a combination are two continuous functions polynomial and a trigonometry function the sign of or the quantity one of Rex is on ly discontinuous, where X is equal to zero. However, that's not included in the domain. So at the moment it is definitely continuous on all riel numbers. The issue is at X equals zero. We have to make sure that this corresponds this f of X equals zero corresponds to the limit. As dis approaches zero s O to confirm, we want to make sure that the limit is experts is zero out of the function except forthe time, Sign of the quantity. One Rex equals F zero, which is here on this case. Andi, Until we confirm that we cannot say that this function is continuous on all real numbers. Once we confirmed this, then we can say it. It is continuous from negative infinity to infinity. So what we do is we approach this using the squeeze Terram. We know that the sine function its value. Its range is from negative one to one. And if we want to buy exit the fourth to each turn, this gives us the function in question that we want. Now we take the entire and equality said here and take the limit as X approaches zero. And if you notice the limited express zero of negative X of the fourth will be zero. The limit as X approaches zero of X to the fourth will be zero. And so this limit of this function has experts. Zero must be between zero and zero, and the only Whether that is true is if dysfunction is lim is also equal to zero. So we have taken care of this limit and shown that it is equal to zero and since it is equal to zero, it is equal to zero confirming and that has X approaches. Zero. This function is continuous at X equals zero, and thus the function is continuous from negative infinity to infinity.

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