1. 2.2 Limits Note 1: The number a may be replaced by ∞ 1 Definition of Limit: Suppose that f(x) becomes arbitrarily close to the number L ( f(x)  L ) as x approaches a ( x  a ). Then we say that the limit of f(x) as x approaches a is L and write

2. 2 Example 1: Estimate the limit of the function as x approaches 2.

3. 3 Example 2: Find the limit Note 2: In general, the limit has nothing to do with the value of the function at a. All we are claiming is that the function approaches L. In the previous example Solution: The function under the limit is not defined at x=1, but this doesn't matter because the definition of limit says that we consider value of x that are close to a, but not equal to a.

4. 4 Provided that x ≠ a, the function can be reduced as follows: Then

5. 5 If the value of the limit coincides with the value of the function, the function is called continuous at this point. Definition of Continuity: A function f is continuous at x=a if f is defined at a and Exercises: Determine if the functions in the above examples are continuous at the given points (points of limits)

6. 6 Example: Find the trigonometric limit Continuous at 0?

7. One-sided limits 7 Definitions : If f(x) becomes arbitrarily close to the number L as x approaches a from the left, then we say that L is the left-sided limit of f(x) as x approaches a and write If f(x) becomes arbitrarily close to the number L as x approaches a from the right, then we say that L is the right-sided limit of f(x) as x approaches a and write

8. Example: Heaviside function 8 Continuous at 0?

9. Exercise 3: Find the limit 9 Solution: We use the fact that 1/x approaches 0 as x increases (or approaches infinity). So, we divide both numerator and denominator by the highest power of x and then use the above limit of 1/x.

10. 10 Theorem: Ifand then A. B. C. D.

11. 11 Three types of limit calculation (see numbered exercises): 1.Limit of a continuous function: just substitute the number. 2.0/0 limit: cancel the common factor that gives 0 in the numerator and denominator and then substitute the number. 3.∞/∞ limit (ratio of polynomials at ∞): divide both numerator and denominator by the highest power of x and use the fact that the limit of the reciprocal function at ∞ is 0.

12. 12 Homework Section 2.2: 11,15,27,31,37,41,49,51.