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  # Limits
  
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  ### Limits
  
  1. 1. Unit 1: Limits<br />Helga Hufflepuff<br />
  2. 2. A limit is<br />
  3. 3. A limit is the y-value of a graph as x approaches from both sides<br />
  4. 4.
  5. 5.
  6. 6. A limit does not exist when the y-value is different as you approach the same x value from different sides<br />
  7. 7. If the function is continuous…<br />you can draw the function without ever having to pick up your pen from the graph<br />The limit value is the same as the function value<br />
  8. 8. Discontinuity<br />Types of Discontinuity:<br />Jump <br />Hole<br />Infinite (Asymptote)<br />
  9. 9. Left- and Right-Hand Limits<br />Sometimes, limits do not approach the same y-value as they approach the same x-value, but the limit still exists<br />If approaching from the left -<br />If approaching from the right +<br />
  10. 10. L’Hopital’s Rule<br />Used to calculate limits involving indeterminate forms; 0/0 or ∞/∞<br />Is there a limit?<br />
  11. 11. 1) Let’s find the derivative of y/x:<br />2) Since x’=1, we can now find the limit.<br /> The limit is 1<br />
  12. 12. The Squeeze Theorem<br />If the limit as h(x) approaches a is equal to the limit as f(x) approaches a—at point L-- and f(x)<g(x)<h(x), then g(x) equals L<br />
  13. 13. Continuous VS. Differentiable<br />Differentiable– Does the derivative exist?<br />If a function is not continuous it cannot be differentiable on all reals<br />Continuous on all reals. Does the derivative exist<br />
  14. 14. Differentiable?<br />If the function is continuous, we only need to worry about where derivative is undefined<br />Cusps <br />Corners<br />At x=2, the derivative is undefined<br />The function is differentiable on all reals except for where x=2<br />
  15. 15. Continuous and<br /> Differentiable<br />
  16. 16. In the Absolute Value function, the derivative is undefined at x=0<br />NOT DIFFERENTIABLE<br />
  17. 17. Intermediate Value Theorem<br />If the function is continuous from [a,b], then there must be a point c in the interval [a,b] and it must have a y-value that is between f(a) and f(b) <br />