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Find the indicated limit.$$\lim _{h \rightarrow 0} \frac{(x+2 h)^{2}-x^{2}}{2 h}$$

$2 x$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Oregon State University

Harvey Mudd College

Idaho State University

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 find the integrated limit off limit edge approaches to zero X plus to edge Also care minus ex hooker Do I did buy to age If we try to substitute a physical to zero in the given function, we have zero upon zero, which is an intermediate form, so we cannot apply the limit directly. Therefore, we have to redefine the function using algebraic techniques. So for this particular problem here, we will use algebraic identities toe to remove the intermediate form. Therefore, we have the given limit. There is a chance to zero a numerator. We have a plus. Behold, so care. So using this identity, we can expand. This expression is extra pair Unless for Ed Suker less for h X minus expert divided by to reach Now limit it projects to zero. Yeah, we can cancel out plus extra care minus X care and we can factor road four edge from the remaining expression. So when we factor or four edge, we have edge well as X a bond to reach Now here we can do some cancelation. It will be canceled. We can cancel is from numerator and denominator on day two times two is for So we have. We meet at approaches to zero to into edge plus eggs. Now we can apply the limit as it approaches to zero. We have to into zero plus eggs. Finally, we have two X, which is the required limit for the given function.

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