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 2 Use 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 discontinuity f(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

24. Worksheet 1

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