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Determine whether $ f'(0) $ exists.

$ f(x) = \left\{ \begin{array}{ll} x \sin \frac{1}{x} & \mbox{if $ x \neq 0 $}\\ 0 & \mbox{if $ x = 0 $} \end{array} \right.$

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02:29

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 7

Derivatives and Rates of Change

Limits

Derivatives

Campbell University

Oregon State University

University of Michigan - Ann Arbor

Idaho State University

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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So in this problem were given this function F of X equals X Sign of one over X. If x is not equal to zero and zero if X equals zero, we were asked to determine F prime and zero. Well, IF F Promise zero exists, then The limit as X approaches zero. Uh F of X -F of zero Over X -0 0 exists. So let's now look at this. Okay, So this is the limit as X goes to zero of X, Sign of one over X minus zero, Because F0 is zero over X. So this is the limit as X goes to zero Of sign of one over X. Okay, well, the problem is As X goes to zero when over X goes to infinity and sign of infinity, oscillates in the interval from -1-1 and does not converge. So by that then the limit as X goes to zero, Sign of one over X does not exist. So F prime zero does not exist.

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