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Find the indicated limit.$$\lim _{h \rightarrow 0} \frac{\sqrt{1+x+h}-\sqrt{1+x}}{h}$$

$\frac{1}{2 \sqrt{1+x}}$

Calculus 1 / AB

Chapter 2

An Introduction to Calculus

Section 3

Limits and Continuity

Derivatives

Missouri State University

Harvey Mudd College

University of Nottingham

Boston College

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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Find limits

in this problem, we have been given a limit. So we had to find integrated limits for the given function, which is limit edge tends to zero. So care route off scared off one plus x plus edge minus scare rolled off one plus X divided by edge. If we substitute a physical to zero in the given function, we have been intermediate form. So we have we get zero upon zero which is an intermediate form, so we cannot substitute. We cannot apply the limit directly, dear, for we have to redefine the function using the algebraic techniques. So for this particular function, we will use the rationalization method to remove this discontinuity off the function, so limit it approaches to zero. So okay, rude off one plus x plus edge minus scared off one plus x bond edge. Multiply by Just take the congregate off the numerator that is one plus x plus edge plus scared off one plus X divided by scheduled off one plus x plus edge plus soup Beirut off one plus x no, we have limit at approaches to zero. A new miniter. We have a minus three times a plus B, which is equal to and Suker minus Be Scared one plus X. Also care divided by scared off one plus x plus edge edge into scare route off one glass X. Now we can cancel this appearance. Care old. So from numerator, we have limit approaches to zero from new military. We have one plus X plus at minus when distribute minus. We have minus X, divided by edge into scared off one bless X plus edge plus scared off one plus X. So from Newman Regular, you can cancel plus X minus, X plus one minus one, and we can also cancel out it from numerator and denominator. So remaining expression we have limit at approaches to zero from new military. We have one upon yeah, scared off one plus X plus Edge, plus Suki route off one plus X So now we can apply the limit is it approaches to zero? We have run up on scare route off one plus EXP NAS zero Yes, scared off one plus x. So finally, we have went up on. So Kuroda one plus x plus Kuroda one plus X becomes too into scared off Reddick Any sense so we can add the coefficients. So ideas two into scared off one plus X, which is required limit or the given function

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