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11.1 An Introduction to Limits Lim f(x) = L as x  a x  a - is as x approaches a from the left x  a + is as x approaches a from the right In order for.

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Presentation on theme: "11.1 An Introduction to Limits Lim f(x) = L as x  a x  a - is as x approaches a from the left x  a + is as x approaches a from the right In order for."— Presentation transcript:

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2 11.1 An Introduction to Limits Lim f(x) = L as x  a x  a - is as x approaches a from the left x  a + is as x approaches a from the right In order for there to be a limit for x  a the limits from the left and right must be the same

3 11.1 An Introduction to Limits Polynomials –Evaluate f(x) for a –Lim (x 2 -3x+4) as x  1 is (1) 2 -3(1)+4 –Create a chart with values at different distances from a Piecewise functions –Same as polynomials

4 11.1 An Introduction to Limits When is there no limit? –When x  a - ≠ x  a + –F(x) becomes infinitely large –F(x) oscillates

5 11.2 Techniques for Calculating Limits Rules for Limits –(Constant) for any constant –(Limit of x) –(Multiple) –(Sum) –(Difference) –(Product) –(Quotient)

6 11.2 Techniques for Calculating Limits Limits (continued) –(Power) –(Polynomial) –(Rational) –(Logarithm) lim[log b f(x)] = log b [lim f(x)] when f(x)>0 & b>0 & b ≠ 1

7 11.3 One-Sided Limits; Limits Involving Infinity if and only if AND Limits at Infinity of 1/x n

8 11.3 One-Sided Limits; Limits Involving Infinity If n>m, the limit does not exist If n=m, the limit is equal to a/b If n

9 11.4 Tangent Lines and Derivatives Slope of the tangent line of y=f(x) at (a, f(a)) has the slope provided the limit as x  a exists If a is in the domain of f, then the derivative of f at a is defined by the previous formula, provided the limit x  a exists.

10 11.4 Tangent Lines and Derivatives Velocity as a derivative –If an object is moving along a straight line and its position on the line at time t is s(t), then the velocity at time a is s’(a) = lim t  a [s(t) – s(a)]/[t-a]

11 11.4 Tangent Lines and Derivatives Geometric definition of f’(a): f’(a) is the slope of the tangent line to f(x) at x=a Algebraic definition of f’(a): Interpretations of f’(a): f’(a) represents velocity when f(x) is a position function and marginal change when f(x) is an economics function.

12 Links o/derplot/ - tangent demohttp://www.math.uri.edu/~bkaskosz/flashm o/derplot/ bin/manager - COWhttp://cow.math.temple.edu/~cow/cgi- bin/manager s/http://archives.math.utk.edu/visual.calculu s/


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