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Limits: The Building Block of Calculus

Limits: The Building Block of Calculus. Steps to finding a limit Limits approaching a value One-Sided limits Limits at infinity Limits with radicals Trig Limits Limits of ln (x) and e x. 1- Steps:. Direct Substitution Outcomes: A number means - continuous

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Limits: The Building Block of Calculus

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  1. Limits: The Building Block of Calculus Steps to finding a limit Limits approaching a value One-Sided limits Limits at infinity Limits with radicals Trig Limits Limits of ln(x) and ex

  2. 1- Steps: • Direct Substitution Outcomes: • A number means - continuous • Numerator and denominator of a rational function both = 0 - not continuous(hole) • Numerator is a real #, but the denominator is 0. (approaches an vertical asymptote)

  3. 2- Limits Approaching a Value Plug in the number Example: x2-9 → (2)2-9=-5 → This equation is a continuous function

  4. The End… Katherine Pistorius Meghan Weisel

  5. 2- Limits Approaching a Value Plug in the number Example 2: (x2-4)/(x-2) → (4-4)/(2-2) =0/0 → Factor Top and Cancel → Plug in 2 → Hole (2,4)

  6. 2- Limits Approaching a Value Plug in the number Example 3: x/(x-1) → 1/(1-1) = 1/0 → Vertical asymptote at x=1 → DNE

  7. 3- One-Sided Limits Used to describe the value of the function as x approaches a specified value from a given direction. Example 1: x/(x-1) →Graph → + Infinity

  8. 4- Limits at Infinity Degree n < Degree d y=0 Degree n > Degree d No asymptote Degree n = Degree d Leading Coefficient (n/d) Example: (2x2-3x+1)/(5x2+2x-3) →leading coefficients are equal → The answer is 2/5

  9. 5- limits with Radicals The limit of an nth root is the nth root of the limit Example: → put the limit inside of the 3rd root → answer:

  10. 6- Trig Limits Example: cos(1/x) cos (1/x) cos(0) → the answer is: 1 → →

  11. 7- Limits of lnx and ex ln(x)= +∞ ln(x) = -∞ ln(x)=DNE ex = +∞ ex = 0

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