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If $ f(x) = \left\{ \begin{array}{ll} x^2 & \mbox{if $ x $ is rational}\\ 0 & \mbox{if $ x $ is irrational} \end{array} \right.$prove that $ \displaystyle \lim_{x \to 0}f(x) = 0 $.

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Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Missouri State University

Oregon State University

University of Nottingham

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 sixty one of this tour Capitalist states Addition Section two point three. It affects equals. This piece was function X squared. If X is irrational, zero effects is irrational. Prove that the Limited's experts zero f is equal to zero. We're going to approach this limit by understanding the F, as you can tell by the parts of it is a function that lines in between zero, which is its minimum and X squared, which is its maximum. So in this case, we will do our best to see if we can use eh squeezed hero to determine the limit as expert to zero for this function and with squeezed here, Um, what we do is we to recall we choose to function the one that is definitely always greater. Um around the region of exit protein A from this case zero. This function is always great enough and then the lower functions always less than half. And then we compare the limits and how they behave as limited has the element of it. Lim approaches zero for this function. Zero. That limit is equal to zero. The limit is export. Zero of this function X squared. It's also is here instant system. It must lie between these two values. Then we have confirmed. But with the squeeze serum, the Ltd's experts zero over their function in it must be equal to Syria.

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