Math Section Bekki George: 01/16/19. University of Houston. Bekki George (UH) Math /16/19 1 / 31

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1 Math 1431 Section Bekki George: University of Houston 01/16/19 Bekki George (UH) Math /16/19 1 / 31

2 Office Hours: Mondays 1-2pm, Tuesdays 2:45-3:30pm (also available by appointment) Office: 218C PGH Course webpage: Bekki George (UH) Math /16/19 2 / 31

3 Section The Idea of a Limit So, what conditions would make a limit not exist? Bekki George (UH) Math /16/19 3 / 31

4 Section The Idea of a Limit Express this limit in words and interpret its meaning: x 2 4 lim x 2 x 2 = 4 Bekki George (UH) Math /16/19 4 / 31

5 Quiz 1 Examples ( ) 5 4. Evaluate the limit: lim x 4 x + 4 Bekki George (UH) Math /16/19 5 / 31

6 Quiz 1 Examples ( 6x 2 ) 7x 5. Evaluate the limit: lim x 0 x Bekki George (UH) Math /16/19 6 / 31

7 Quiz 1 Examples 7. Evaluate the limit: lim f (x). Given that x 4 { 4 x x < 4 f(x) = 16 x > 4 Bekki George (UH) Math /16/19 7 / 31

8 Quiz 1 Examples 10. Evaluate the limit: lim f (x). Given that x 2 + { 4 x + 2 x 2 f(x) = x 2 x x > 2 Bekki George (UH) Math /16/19 8 / 31

9 Suppose we want to find the distance between two numbers a and b on the number line. How do we represent this distance mathematically? Bekki George (UH) Math /16/19 9 / 31

10 When we discuss limits, we say we want x to be arbitrarily close to a but it doesn t have to equal a. How can we represent this mathematically? Bekki George (UH) Math /16/19 10 / 31

11 Once we are close enough to x, we find our limit by looking at what our y values are close to. (Remember, y = f(x)). Let L represent our answer for the limit and let ɛ be our distance that is close enough. In mathematical terms, we will write this as: Bekki George (UH) Math /16/19 11 / 31

12 The formal definition of a limit Let f be a function defined on the intervals (c δ, c) and (c, c + δ), where δ > 0. lim f(x) = L x c if and only if for each ɛ > 0, there exists a δ > 0 such that if then 0 < x c < δ f(x) L < ɛ Bekki George (UH) Math /16/19 12 / 31

13 Bekki George (UH) Math /16/19 13 / 31

14 Bekki George (UH) Math /16/19 14 / 31

15 Example: Show that lim x 2 (3x + 5) = 11 using the definition of limit. Bekki George (UH) Math /16/19 15 / 31

16 Example: Give the largest δ that works with ɛ = 0.1 for the limit lim (1 2x) = 3 x 1 Bekki George (UH) Math /16/19 16 / 31

17 Example: Give the largest δ that works with ɛ = 0.1 for the limit lim(4x 5) = 7 x 3 Bekki George (UH) Math /16/19 17 / 31

18 You try: Give the largest δ that works with ɛ = 0.02 for the limit lim (2x + 5) = 3 x 1 Bekki George (UH) Math /16/19 18 / 31

19 Formal definitions for Left-handed and Right-handed limits: Let f be a function defined on the interval (c δ, c), where δ > 0. lim f(x) = L x c if and only if for each ɛ > 0, there exists a δ > 0 such that if c δ < x < c then f(x) L < ɛ. Let f be a function defined on the interval (c, c + δ), where δ > 0. lim f(x) = L x c+ if and only if for each ɛ > 0, there exists a δ > 0 such that if c < x < c + δ then f(x) L < ɛ. Bekki George (UH) Math /16/19 19 / 31

20 Uniqueness of a limit If lim x c f(x) = L and lim x c f(x) = M then L = M. The limit of a sum: If lim f(x) = L and lim g(x) = M then lim(f(x) + g(x)) = L + M x c x c x c (provided each limit exists). The limit of a difference: If lim f(x) = L and lim g(x) = M then lim(f(x) g(x)) = L M x c x c x c (provided each limit exists). Bekki George (UH) Math /16/19 20 / 31

21 The limit of a product: If lim f(x) = L and lim g(x) = M then lim(f(x)g(x)) = L M x c x c x c (provided each limit exists). The limit of a quotient: f(x) If lim f(x) = L and lim g(x) = M with M 0, then lim x c x c x c g(x) = L M (provided each limit exists). Bekki George (UH) Math /16/19 21 / 31

22 Note: If lim f(x) = L and lim g(x) = M with L 0 and M = 0, then x c x c f(x) lim does not exist. x c g(x) Bekki George (UH) Math /16/19 22 / 31

23 Graph f(x) = 2 What is lim x 4 f(x)? What is lim x 1 f(x)? What is lim x 0 f(x)? So, the limit of a constant function is that function: If f(x) = k then lim x a k = k Bekki George (UH) Math /16/19 23 / 31

24 Graph f(x) = x What is lim x 4 f(x)? What is lim x 1 f(x)? What is lim x 0 f(x)? So: If f(x) = x then lim x a x = a Bekki George (UH) Math /16/19 24 / 31

25 What is a polynomial function? A polynomial function is any function of the form: f(x) = a n x n + a n 1 x n a 1 x + a 0 Where a k is a real number and n is an integer. Examples: f(x) = x 2 4 g(x) = 1 3 x5 + 2x 3 x + 1 h(x) = 3x Bekki George (UH) Math /16/19 25 / 31

26 Because a polynomial function is just a combination of linear functions and constants and we know that we can find the limit for linear and constant functions, we can easily find the limit of polynomial functions. So, to find lim P (x), where P (x) is a polynomial function, just plug in x a a. In other words, lim P (x) = P (a) x a Bekki George (UH) Math /16/19 26 / 31

27 What is a rational function? A rational function, R(x) is the quotient of two polynomial functions: R(x) = P (x) Q(x) Recall, The limit of a quotient: f(x) If lim f(x) = L and lim g(x) = M with M 0, then lim x c x c x c g(x) = L M So, providing lim x c Q(x) 0, we can just plug in to find the answer for a limit of a rational function. However, if lim x c Q(x) = 0, we need to consider other things. Bekki George (UH) Math /16/19 27 / 31

28 Let P (x) and Q(x) be polynomial functions and let a be a real number. Then, { P (a) P (x) lim x a Q(x) = Q(a) if Q(a) 0 undefined if P (a) 0 and Q(a) = 0 If P (a) and Q(a) both equal 0 then more work is required. Bekki George (UH) Math /16/19 28 / 31

29 Techniques to evaluate limits: direct substitution cancellation rationalization algebraic simplification Bekki George (UH) Math /16/19 29 / 31

30 Examples: 1 lim x 2 (3x2 + 1) = 2 lim x 0 x 2 2x 2x + 1 = 2x 2 x 3 3 lim = x 1 x + 1 Bekki George (UH) Math /16/19 30 / 31

31 To Do Read your syllabus and take the CPQ. Sign up for and take the Diagnostic Exam. Read Sections 1.2 and 1.3. Take quiz 1 & 2. me questions if you have any. Bekki George (UH) Math /16/19 31 / 31

Math Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37

Math Section Bekki George: 08/28/18. University of Houston. Bekki George (UH) Math /28/18 1 / 37 Math 1431 Section 14616 Bekki George: bekki@math.uh.edu University of Houston 08/28/18 Bekki George (UH) Math 1431 08/28/18 1 / 37 Office Hours: Tuesdays and Thursdays 12:30-2pm (also available by appointment)