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Limits - Substitution

Limits - Substitution. As x approaches 3 from both directions, y approaches 8. We can find the limit by substituting x = 3 into the equation. Practice. Answer: The limit is 24.

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Limits - Substitution

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  1. Limits - Substitution

  2. As x approaches 3 from both directions, y approaches 8 We can find the limit by substituting x = 3 into the equation

  3. Practice Answer: The limit is 24

  4. When we try to substitute into we get which is undefined. If we draw the graph we find that we get a straight line with equation

  5. The hole in the graph at x = 1 is a discontinuity.y has a value for every x except x = 1. i.e.

  6. You can recognise a discontinuity because you need to lift your pen to continue your graph. The graph below is continuous because we can draw it without having to lift the pen.

  7. Although , we do have a limit at x = 1.

  8. Two methods to find the limit.Method 1 Now substitute x = 1 to get a limit of 2 i.e.

  9. Method 2Use L’Hospital’s Rule Note: Only use this when substitution gives 0/0

  10. Practice Answer: Substituting gives Using either factorising or L’Hospital’s Rule: Limit is

  11. Discontinuities and limits

  12. f(0.5) = 3 (Solid dot gives the value at 0.5)

  13. But

  14. Not all discontinuities have a limit

  15. Jump discontinuity

  16. The graph is not heading towards the same value so there is no limit. Tends towards 1 Tends towards -1

  17. Limit at x = -4 does not exist

  18. Vertical asymptotoes: Limit does not exist.

  19. Note: this is NOT a discontinuityf(1) =2 and

  20. More Limits Divide top and bottom by x

  21. When bottom power is greater than top power

  22. When top power is greater than bottom power

  23. Worksheet 1

  24. What do you think the limit is at x = 0?

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