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Definition: Let f be a function of two variables defined for all points "near" (a,b) but possibly not defined at the point (a,b). We say the . limit of f(x,y) as (x,y) approaches (a,b) is L, written as. lim f(x,y) = L , (x,y)!(a,b) if the distance from f(x,y) to L, | f(x,y) – L | , can be made arbitrarily .
Chapter 4 Limits and Continuity: Exercises. (Updated solution) 1. Given the graph of the function g(t) , nd. 1. lim g(t) = [-1] t!0. 5. lim g(t) = [2] t!2. 2. lim g(t) = [-2] t!0+. 3. lim g(t) t!0. = [does not exist] 6. lim g(t) = [0] t!2+. 7. lim g(t) t!2. = [does not exist] 8. g(2) = [1] 4. g(0) = [-1] 9. lim g(t) t!4. = [3] 2.
The correct and most appropriate answer to a multiple-choice question will be, in each case, just one of the seven choices (A), (B), (C), (D), (E), (F), or (G). Answer all multiple-choice questions on the answer sheet, which is page 14 of this exam.
• We will use limits to analyze asymptotic behaviors of functions and their graphs. • Limits will be formally defined near the end of the chapter. • Continuity of a function (at a point and on an interval) will be defined using limits.
Limits are a very powerful tool in mathematics and are used throughout calculus and beyond. The key idea is that a limit is what I like to call a \behavior operator". A limit will tell you the behavior of a function nearby a point.
LIMITS BY STANDARD EXPANSIONS. Write down the first two non zero terms in the expansions of sin3x and cos2x . Hence find the exact value of. 3 x cos2 x − sin3 x . lim 3 . x → 0 3 x . sin3 x ≈ 3 x − 9 x 3 , cos2 x ≈ 1 − 2 x 2 , − 1. 2 2. Use standard expansions of functions to find the value of the following limit.
4.3 Rates of Change in Applied Contexts Other Than Motion. 4.6 Approximating Values of a Function Using Local Linearity and Linearization. 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms. 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points.
