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Exploring Limits and Discontinuities in Functions

This text discusses the concept of limits in calculus, particularly as a variable approaches specific values. It illustrates how to evaluate limits via substitution, understanding discontinuities, and applying L'Hospital's Rule when applicable. Key concepts include recognizing when a function's graph has discontinuities, such as jump discontinuities or vertical asymptotes. Furthermore, it provides examples and practice questions to reinforce understanding of these mathematical concepts, making it easier for students to grasp limits and their implications in function behavior.

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Exploring Limits and Discontinuities in Functions

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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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