Explanation Given: The given limit is lim x → 0 sin − 1 x x − 1 Concept Used: Basic properties and rules for limits: for any real number c, suppose that functions f and g has limit at x = c , then Constant rule: lim x → c k = k for any constant k Multiple rule: lim x → c k f ( x ) = k lim x → c f ( x ) for any constant k Sum rule : lim x → c ( f ( x ) + g ( x ) ) = lim x → c f ( x ) + lim x → c g ( x ) Power rule: lim x → c ( f ( x ) ) n = ( lim x → c f ( x ) ) n where n is a rational number Product rule: lim x → c [ f ( x ) g ( x ) ] = ( lim x → c f ( x ) ) ( lim x → c g ( x ) ) Quotient rule: lim x → c [ f ( x ) g ( x ) ] = lim x → c f ( x ) lim x → c g ( x ) if lim x → c g ( x ) ≠

7th Edition

ISBN: 9781524916817

Author: SMITH KARL J, STRAUSS MONTY J, TODA MAGDALENA DANIELE

Publisher: Kendall Hunt Publishing

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topic…

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Chapter 2.2, Problem 3PS

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To evaluate the given limit

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Evaluate the following integrals, showing all your working

Consider the function f(x) = 2x³-4x2-x+1. (a) Without doing a sketch, show that the cubic equation has at least one solution on the interval [0,1]. Use a theorem discussed in lectures, or see Section 1.8 of Calculus (7th ed) by Stewart. Ensure that the conditions of the theorem are satisfied (include this in your solution) (b) Now, by sketching the cubic (by hand or by computer), you should see that there is, in fact, exactly one zero in the interval [0,1]. Use Newton's method to find this zero accurate to 3 decimal places. You should include a sketch of the cubic, Newton's iteration formula, and the list of iterates. [Use a computer if possible, e.g., a spreadsheet or MatLab.]

Evaluate the following integrals, showing all your working

Chapter 2 Solutions

Calculus

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To evaluate the given limit

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