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Lecture 2_2Ma141

Lecture 2_2Ma141 - Picture on the board Right-hand limit...

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Math 141 Section 2.2 Limit of a function Outline of lecture I. Define : The limit of a function (refer to text p. 99) lim x a f ( x ) = L Graph on board: The values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) but x not equal to a. Three “pictures” to keep in mind with this definition, all of these will have the limit existing and being L a) a smooth continuous function with f(a)=L b) a function that is discontinuous at x=a and f(a) a L c) a function that is not defined at x=a (hole in the graph) Examples i) lim x 2 x 2 + 5 x +1
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ii) lim x 4 x 2 - 16 x - 4 iii) lim x 0 sin( 1 x ) iv) lim x 0 | x | x II. In the last example we see that there is not ONE limit but there does seem to be “limiting behavior” on either side of the x-value we are looking at. Define: One-sided Limits
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Unformatted text preview: Picture on the board: Right-hand limit: lim x a + f ( x ) = L As x approaches a from the right the associated f(x) values approach L. Right hand Limit: lim x a-f ( x ) = M As x approaches a from the left, the associated f(x) values approach M. Note: For the THE limit to exist the limit to exist at x=a, the left and right limit must be the same. Example #4 p.106 in the text Example: #6 p. 106 sketch the following piece-wise defined graph and use your graph to determine for what values of a does the limit exist? Example: #24 p.107 The slope of the tangent line to the graph of the exponential function y = 2 x at the point (0,1) is lim x 2 x- 1 x , (make sure you see how they got this) . Estimate this slope to 3 decimal places....
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