What is “calculus”? What do you learn in a calculus class? How do algebra and calculus differ?  You will be able to answer all of these questions after you finish the course.

10.1 Introduction to Limits

One of the basic concepts to the study of calculus is the concept of limit. This concept will help to describe the behavior of f(x) when x is approaching a particular value c. In this section, we will review and learn more about functions, graphs, and limits

When x is closer and closer to 2, F(x) is closer to 3 Therefore: Example 1a: f(x) = 2x - 1 Discuss the behavior of of f(x) when x gets closer to 2 using graph Graph f(x) = 2x - 1 When x is closer and closer to 2, F(x) is closer to 3 Therefore: The limit of f(x) as x approaches 2 is 3 lim(2x-1) = 3 = f(2) X2

Example 1b: f(x) = 2x - 1 Discuss the behavior of the values of f(x) when x gets closer to 2 using table

Find: lim (x+2) and lim (3x+1) Do you get 2 and -2? If not, try again

Example 2: Discuss the behavior of f(x) when x gets closer to 2 If x = 2, f(x) is undefined. If you graph, you will see a hole there. x 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 f (x) 3.5 3.9 3.99 3.999 ? 4.001 4.01 4.1 4.5 Therefore, when x is closer and closer to 2, f(x) is closer to 4 lim f(x) = 4 = f(2) or X2

Example 2: Discuss the behavior of the values of f(x) when x is closer to 2. Does the limit exist? 1 1.9 1.99 2 2.001 2.01 2.1 2.5 f (x) -1 ? * This function is not defined when x = 2. * The limit does not exist because the limit on the left and the limit on the right are not the same. Lim f(x) = -1 represents the limit on the left of 2 Lim f(x) = 1 represents the limit on the right of 2 X2 - X2 +

We write and call K the limit from the left (or left-hand limit) if f (x) is close to K whenever x is close to c, but to the left of c on the real number line. and call L the limit from the right (or right-hand limit) if f (x) is close to L whenever x is close to c, but to the right of c on the real number line. In order for a limit to exist, the limit from the left and the limit from the right must exist and be equal.

Example 3 Discuss the behavior of f(x) for x near 0 Lim f(x) = F(0) = X0 - X 0 + X0

Example 3 - continue (B) Discuss the behavior of f(x) for x near 1 Lim f(x) = F(1) = 1 X1 - 2 X 1 + does not exist X1 not defined

Example 3 - continue (C) Discuss the behavior of f(x) for x near 3 Lim f(x) = F(3) = 3 X3 - 3 X 3 + 3 X3 not defined

If you don’t get -4, try again Example 4 Lim (x2 + 5x) = lim x2 + lim 5x = (lim x) (lim x) + 5 (lim x) = (-2) (-2) + 5 (-2) = 4 – 10 = -6 X-2 Property 1 X-2 X-2 Property 4 and 3 X-2 X-2 X-2 Try lim (x4 + 3x – 2) If you don’t get -4, try again X-1

If you don’t get 2, try again Example 5 Lim = X2 Property 8 X2 If you don’t get 2, try again Try lim X-1

If you don’t get 1/3, try again Example 6 Note that this is a rational function with a nonzero denominator at x = -2 = = If you don’t get 1/3, try again

Example 7 Lim f(x) = lim (2x+3) = 2(5)+ 3 = 13 B) Lim f(x) = If x < 5 If x > 5 Lim f(x) = lim (2x+3) = 2(5)+ 3 = 13 X5- X5- B) Lim f(x) = lim (-x+12)=-5+12=7 X5+ X5+ C) Lim f(x) = Does not exist because the left hand the right hand limits are not equal X5 D) F(5) = is not defined

Example 8: Use algebraic and/or graphical techniques to analyze each of the following indeterminate forms A) B) C) See next page for step by step instruction

Example 8 - Solutions Note: when you find the limits of the above problems, you must factor first and then simplify before you substitute the number for x

Different Quotient (pre-cal) Indeterminate form

See the next pages for step by step instruction Examples: Find the following limit for the following functions 9) 10) 11) See the next pages for step by step instruction

Because the limit on the left and the litmit on the right are not the same. Therefore, this limit does not exist

Rationalize the numerator