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Chapter 2 Limit and continuity <ul><li>Tangent lines and length of the curve </li></ul><ul><li>The concept of limit </li></ul><ul><li>Computation of limit </li></ul><ul><li>Continuity </li></ul><ul><li>Limit involving infinity (asymptotes </li></ul>
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1)Tangent lines and the length of the curve <ul><li>Tangent line </li></ul>
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<ul><li>Lets zoom the graph </li></ul><ul><li>Now we can see that the slope get closer To get the correct tangent line of the graph at any point, we need to zoom the graph as can as possible </li></ul>
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2)The concept of limit <ul><li>Example 1 </li></ul><ul><li>f is not defined at x=2 </li></ul><ul><li>Example 2 </li></ul><ul><li>g is not defined at x=2 </li></ul>
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Thus, we can conclude that the limit of f(x) when x approaches 2 is 4 and we write <ul><li>We can see that when x get closer to 2 from left side, f(x) get closer to 4 </li></ul>We can see also that when x get closer to 2 from right side, f(x) get closer to 4 3.9999 1.9999 3.999 1.999 3.99 1.99 3.9 1.9 4.0001 2.0001 4.001 2.001 4.01 2.01 4.1 2.1
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Since the limits from the two sides are different, we conclude that the limit of g(x) when x approaches 2 doesn’t exist <ul><li>We can see that when x get closer to 2 from left side, g(x) increase very fast toward </li></ul>We can see also that when x get closer to 2 from right side,g(x) decrease very fast toward 10003.9999 1.9999 1003.999 1.999 103.99 1.99 13.9 1.9 -9995.9999 2.0001 -995.999 2.001 -95.99 2.01 -5.9 2.1
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<ul><li>Def: </li></ul><ul><li>A limit exist if and only if the two one-sided limit exist and are equal </li></ul><ul><li>for some L, if and only if </li></ul><ul><li>Exercise: </li></ul><ul><li>Find out why exist </li></ul><ul><li>while does not exist </li></ul><ul><li>Example: </li></ul><ul><li>Evaluate the below limit is it exist </li></ul>
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3)Computation of the limit <ul><li>Theorem1: </li></ul><ul><li>Example evaluate the following limits </li></ul><ul><li>Theorem2: </li></ul><ul><li>For any Polynomial p(x) and any real number a, </li></ul><ul><li>Theorem 3: </li></ul><ul><li>suppose that , then </li></ul><ul><li>Example: evaluate </li></ul>
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<ul><li>Theorem 4: </li></ul><ul><li>Example: evaluate the following limit </li></ul><ul><li>Theorem 5 (squeeze theorem) </li></ul><ul><li>Suppose that </li></ul><ul><li>for all x in some interval and </li></ul><ul><li>for some number L, then also </li></ul><ul><li>Example: evaluate the limit </li></ul><ul><li>Example : ( a limit of piecewise-defined function) </li></ul>
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4)Continuity of Functions <ul><li>Definition </li></ul><ul><li>A function f is continuous at a point x = a if </li></ul><ul><li>is defined </li></ul>
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Problem 14 Answer Removable Removable Not removable
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<ul><li>Theorem 1 </li></ul><ul><li>All Polynomial </li></ul><ul><li>are continuous everywhere, is continuous for all x if n is odd, and </li></ul><ul><li>continuous for all x 0 if n is even. Also is continuous for all , and for </li></ul><ul><li>Theorem 3 </li></ul><ul><li>Suppose that and is continuous at , then </li></ul><ul><li>Theorem 2 </li></ul><ul><li>Suppose that f(x) and g(x) are continuous at x=a, then </li></ul><ul><li>(f+g)(x) is continuous at x=a </li></ul><ul><li>(f-g)(x) is continuous at x=a </li></ul><ul><li>(f.g)(x) is continuous at x=a </li></ul><ul><li>(f/g)(x) is continuous at x=a if </li></ul><ul><li>g(a) 0 </li></ul><ul><li>Example </li></ul><ul><li>Determine where is continuous? </li></ul><ul><li>Corollary </li></ul><ul><li>Suppose that is continuous at and is continuous at , then </li></ul><ul><li>is continuous at </li></ul>
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<ul><li>Practice on continuity </li></ul><ul><li>Exercise 11 : explain why each function is discontinuous at the given point by indicating which of the three condition in definition are not met </li></ul><ul><li>Exercise 23,19 : find all discontinuity of f(x). If the discontinuity is removable, introduce the new function that remove the discontinuity: </li></ul><ul><li>Exercise 34 : determine the value of a and b that make the given function continuous </li></ul>
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<ul><li>Theorem4 : (intermediate value theorem) </li></ul><ul><li>Suppose that is continuous on closed interval [a, b], and W is any number between f(a) and f(b). Then, there is a number </li></ul><ul><li>For which </li></ul><ul><li>corollary2 : Suppose that f(x) is continuous on closed interval (a, b), and f(a) and f(b) have opposite signs (f(a).f(b)<0), so there is at least one number </li></ul><ul><li>for which f(c)=0 </li></ul>