1. Solved Problems on Limits and Continuity

2. Overview of Problems 1 2 4 3 5 6 7 8 9 10

3. Overview of Problems 11 12 13 14 15

4. Main Methods of Limit Computations1
The following undefined quantities cause problems: 2
In the evaluation of expressions, use the rules
If the function, for which the limit needs to be computed, is defined by an algebraic expression, which takes a finite value at the limit point, then this finite value is the limit value. 3
If the function, for which the limit needs to be computed, cannot be evaluated at the limit point (i.e. the value is an undefined expression like in (1)), then find a rewriting of the function to a form which can be evaluated at the limit point. 4

5. Main Computation Methods
Frequently needed rule 1
Cancel out common factors of rational functions. 2
If a square root appears in the expression, then multiply and divide by the conjugate of the square root expression. 3
Use the fact that 4

6. Continuity of Functions
Functions defined by algebraic or elementary expressions involving polynomials, rational functions, trigonometric functions, exponential functions or their inverses are continuous at points where they take a finite well defined value. 1
A function f is continuous at a point x = a if 2
Used to show that equations have solutions.
The following are not continuous x = 0: 3 4
Intermediate Value Theorem for Continuous Functions
If f is continuous, f(a) < 0 and f(b) > 0, then there is a point c between a and b so that f(c) = 0.

7. Limits by Rewriting
Problem 1
Solution

8. Limits by Rewriting
Problem 2
Solution

9. Limits by Rewriting
Problem 3
Solution
Rewrite

10. Limits by Rewriting
Problem 4
Solution
Rewrite
Next divide by x.

11. Limits by Rewriting
Problem 5
Solution
Rewrite
Next divide by x.

12. Limits by Rewriting
Problem 6
Solution

13. Limits by Rewriting
Problem 7
Solution
Rewrite:

14. Limits by Rewriting
Problem 8
Solution
Rewrite:

15. Limits by Rewriting
Problem 9
Solution
Rewrite

16. Limits by Rewriting Problem 9 Solution (cont’d) Rewrite
Next divide by x.
Here we used the fact that all sin(x)/x terms approach 1 as x  0.

17. One-sided Limits
Problem 10
Solution

18. Continuity
Problem 11
Solution

19. Continuity
Problem 12
Solution

20. Continuity
Problem 13
Solution

21. Continuity
Problem 14
Answer
Removable
Not removable

22. Continuity Problem 15 Solution
By the intermediate Value Theorem, a continuous function takes any value between any two of its values. I.e. it suffices to show that the function f changes its sign infinitely often.