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Discuss (i) the continuity and; (ii) the differentiability of the given function. (iii) If a discontinuity is removable, redefine the function so as to make it continuous there. (iv) Is it now differentiable?$$\text { (a) } f(x)=\frac{x^{2}-9}{x-3} \quad \text { (b) } g(x)=\left\{\begin{array}{cl}x+3 & \text { if } x \neq 3 \\4 & \text { if } x=3\end{array} \quad(\mathrm{c}) \mathrm{h}(\mathrm{x})=\mathrm{x}+3\right.$$

(a) and(b) (i) disc at $x=3$ (ii) diff. everywhere except $x=3$ (iii) $f(3)=6$ (iv) yes(c) everywhere cont. and diff.

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.

05:02

Determine whether or $\ope…

here we have to find whether this for ransom is continuous. A tax is called 2 3 or differentiable attacks is called 2, 3. So for continuity X Squire minus three at X is equal to three MS three Squire -3 should be equal to exposed to it accessible to three. Let's do no. Terry, Squire is nine minus three And this is five. Six is not equal to five. It means that the function it is north not continuous at accessible to did he? Since it is not continuous, it is also not differences differentiable at Access code 2 3? No, check for the left. All right, left, continued continuity and right continuity since X quiet minus three. If you plug three here, it will be limit extends to zero, X squared minus three. It will be from zero and f of f of trees which is not equal to F of three. That is five F of three will be five. No, Here it is three, not 0 limit Exchange 2, 3 plus X plus two Is equal to five. That is equal to Air Force three. Therefore we can see the function is right continuous. There is there is a chance that johnson maybe right differentiable. No africa's of today from right you can find limit extends to judge ito plus F of three plus edge- Air Force three over H a little bit which tends to zero plus f of three plus H will be three plus H plus two minus f of threes five over the edge, 3-plus 2 is five and 5 -5 will be zero. So here it will be limited, Extends to zero plus Edge over the edge that is equal to one. So the right, the right differential ability exists. It is differentiable from the right, it is continuous from right and it is differentiable from right.

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