Concept explainers
Evaluating a Limit In Exercises 43–60, (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L’Hôpital’s Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b).
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Calculus
- (Right and Left Limits). Introductory calculus coursestypically refer to the right-hand limit of a function as the limit obtained by“letting x approach a from the right-hand side.” (a) Give a proper definition in the style of Definition 4.2.1 ((Functional Limit).for the right-hand and left-hand limit statements: limx→a+f(x) = L and limx→a−f(x) = M. (b) Prove that limx→a f(x) = L if and only if both the right and left-handlimits equal L.arrow_forwardLimit and Continuity In Exercises , find the limit (if it exists) and discuss the continuity of the function. 14. lim (x, y)→(1, 1) (xy) /(x^2 − y^2 ) 16. lim (x, y)→(0, 0) (x^2 y) /(x^4 + y^2)arrow_forwardPrecise Definition of Limit In Exercises 7–10, use the formal definition of limit to prove that the function is continuous at c.arrow_forward
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- Calculus 3 Functions of Several Variables; Limits and Continuity in Higher Dimensions Question 3: Read Example 5 and the boxed text “Two-Path Test for Nonexistence of a Limit” (p. 818 – 819). Explain what the two-path test says and how this shows that the limit in this example does not exist at the origin. Include the details involved in this particular example.arrow_forwardCalculus Given lim (x,y)→(1,2) of (x + y − 3) / (x^2 − 1) Evaluate the limit along the paths y = 2 and y = x + 1. Does the limit exist?arrow_forwardNonexistence of limits Use the Two-Path Test to prove that the following limits do not exist.arrow_forward
- Functions and Change: A Modeling Approach to Coll...AlgebraISBN:9781337111348Author:Bruce Crauder, Benny Evans, Alan NoellPublisher:Cengage Learning