The Tangent Line Problem What is the slope of the black line? As the red point approaches the black point, the red secant line approaches the black tangent line, and The slope of the secant line approaches the slope of the tangent line.

A Geometric Example of limit Look at a polygon inscribed in a circle As the number of sides of the polygon increases, the polygon is getting closer to becoming a circle. n is the number of sides we can make some mathematical statements: As n gets larger, then the red polygon gets closer to being a circle As n approaches infinity, the polygon approaches the circle The limit of the polygon , as n goes to infinity is the circle

Observing a Limit Can be observed on a graph. View Demo 3

Observing a Limit Can be observed on a graph.

Non Existent Limits f(x) grows without bound View Demo3

Non Existent Limits View Demo 4

Formal Definition of a Limit The For any ε (as close as you want to get to L) There exists a  (we can get as close as necessary to c ) •

Formal Definition of a Limit For any  (as close as you want to get to L) There exists a  (we can get as close as necessary to c Such that …

Specified Epsilon, Required Delta

Finding the Required  Consider showing |f(x) – L| = |2x – 7 – 1| = |2x – 8| <  We seek a  such that when |x – 4| <  |2x – 8|<  for any  we choose It can be seen that the  we need is

Example of limits at infinity The function can diverge The function doesn’t converge to a single value but its amplitude keeps growing. It diverges.

Examples of limits at x=0 The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)

Try These Do not use your calculator

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