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Sketch the graph of the function and use it to determine the values of $ a $ for which $ \displaystyle \lim_{x\to a}f(x) $ exists.$ f(x) = \left\{ \begin{array}{ll} 1 + \sin x & \mbox{if $ x < 0 $}\\ \cos x & \mbox{if $ 0 \le x \le \pi $}\\ \sin x & \mbox{if $ x > \pi $} \end{array} \right.$
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05:40
Daniel Jaimes
06:06
Anjali Kurse
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 2
The Limit of a Function
Limits
Derivatives
Ali L.
October 12, 2020
Campbell University
University of Nottingham
Idaho State University
Boston College
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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Okay here we have a photograph of the piecewise defined function F of X one plus one plus sine of X. When X is less than zero. F of X is defined to be co sign of X for X between zero and pi inclusive, and F of X equals sine of X. When X is greater than high. Looking at the graph of F of X, we can see that it is continuous everywhere, except at X equals pi. Uh, So the limit of F of X as X approaches some number A will exist for every number A except high.
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