Calculus I (MAT 145) Dr. Day Monday January 28, 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! Conjugates Common Denominators Expand, Factor, Reduce Quiz #2/3 today! Date Change for Test #1: Friday, Feb 8 Monday, January 28, 2019 MAT 145

Monday, January 28, 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). Monday, January 28, 2019 MAT 145

Monday, January 28, 2019 MAT 145

Monday, January 28, 2019 MAT 145

Vertical Asymptotes Monday, January 28, 2019 MAT 145

Limits and Vertical Asymptotes Use limit statements to describe the vertical asymptotes. Monday, January 28, 2019 MAT 145

LIMIT LAWS! Suppose that c is a constant and the limits and exist. Monday, January 28, 2019 MAT 145

MORE LIMIT LAWS! Monday, January 28, 2019 MAT 145

Monday, January 28, 2019 MAT 145

Monday, January 28, 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: Monday, January 28, 2019 MAT 145

Monday, January 28, 2019 MAT 145

Monday, January 28, 2019 MAT 145

Monday, January 28, 2019 MAT 145

Monday, January 28, 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 . Monday, January 28, 2019 MAT 145