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Determine (a) $\lim _{x \rightarrow 0^{+}} \frac{\left|x^{3}\right|}{x}$ (b) $\lim _{x \rightarrow 0^{-}} \frac{\left|x^{3}\right|}{x}.$

(a) 0(b) 0

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

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Lectures

04:40

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.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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in this problem, we have to determine in airport limited X approaches to zero from right hand side off the function, which is absolute value off X cubed, divided by X. If you substitute accessible to zero, we have an intermediate form, therefore, before going to apply the limit to let us really find the function. So X approaches toe zero from right hand side since this is a right and limit. So we will take the post to value off that stool function, which will be X cube upon eggs so this will be equal to limit off X approaches to zero. From right hand side. We have ex care. Now we can apply the limit. That is, zero scare is equal to zero. In part be off this problem. We have to find the limit X approaches to zero from left hand side off the function absolute value off X Cuba bond X Here again, we cannot apply the limit directly because we we have an intermediate form. Therefore, let us really find the function using algebraic techniques. Since we have a left hand side limit their therefore we will take the negative value off the absolute function divided by eggs. So this will be equal to limit X approaches to zero from left hand side minus ex care. So X can be canceled from numerator will be reduce the exponents off. The numerator will be will become to so we have minus X a care. Now we can apply the limit as X approaches to little from LePen side. Who's to care will be equal to zero hands. Both the limits from left hand side and right hand side are equal for the given function.

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