Calculus I (MAT 145) Dr. Day Wednesday January 23, 2019 Approachment  Limits! Techniques for Determining the Value of a Limit Numerical Approximation: Same process we used last week Graphical Approximation: Visual determination Algebraic Representation: Determining an exact value! Quiz #2 today Friday: Quiz #3 Friday, Feb 1: Test #1 in STV 346

Approachment Friday, January 18, 2019 MAT 145

The concept of . . . LIMIT! This says that the values of f(x) tend to get closer and closer to the number L as x gets closer and closer to the number a (from either side of a) regardless of the value of f(a). Friday, January 18, 2019 MAT 145

“The limit of f(x), as x approaches a, is L.” The concept of . . . LIMIT! “The limit of f(x), as x approaches a, is L.” Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

What happens when you evaluate f(t) at t = 0? Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

t f(t) -0.5 0.165525061 -0.1 0.166620396 -0.01 0.166666204 -0.001 0.166666662 -0.0001 0.16666668 -0.00001 ????? 0.00001 0.0001 0.001 0.01 0.1 0.5 Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

lim 𝑥→1 𝑓(𝑥 Numerical Approachment? Graphical Approachment? Directly? Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

lim 𝑥→1 𝑓(𝑥 Numerical Approachment? Graphical Approachment? Direct Calculation? Friday, January 18, 2019 MAT 145

Write 3 limit statements for this situation. Friday, January 18, 2019 MAT 145

the function exists at a. When limit exists If and , then If the left-hand limit exists and the right-hand limit exists and they are the same value, then the overall (two-sided) limit exists. Limits can exist whether or not the function exists at a. And, if the function does exist, but is not the same value as the limit at a (i.e., f(a)≠L), that’s okay, and, in fact, that’s how we know that a function is discontinuous at x=a (stay tuned: Section 2.6). Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Evaluate each limit, if it exists, based on the graph of g(x). Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Vertical Asymptotes Friday, January 18, 2019 MAT 145

Limits and Vertical Asymptotes Use limit statements to describe the vertical asymptotes. Friday, January 18, 2019 MAT 145

LIMIT LAWS! Suppose that c is a constant and the limits and exist. Friday, January 18, 2019 MAT 145

MORE LIMIT LAWS! Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

What happens when you evaluate f(t) at t = 0? Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

t f(t) -0.5 0.165525061 -0.1 0.166620396 -0.01 0.166666204 -0.001 0.166666662 -0.0001 0.16666668 -0.00001 ????? 0.00001 0.0001 0.001 0.01 0.1 0.5 Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

of the curve y = f(x) if either: The line y = L is called a horizontal asymptote of the curve y = f(x) if either: Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

Friday, January 18, 2019 MAT 145

by finding its intercepts and its limits as and as . Sketch the graph of by finding its intercepts and its limits as and as . Friday, January 18, 2019 MAT 145