Explanation A limit of a function at a point a is defined as the value that the function approaches as its argument approaches point a . The notation of the limit is as follows: lim x → a ( f ( x ) ) = L , which is read as “limit of f ( x ) as x approaches a is L ” One-Sided Limits: A one-sided limit is a limit that only takes into account the values of a function that approach a value from only one side i.e. either above or below. The right side limit of a function f as it approaches a is give as lim x → a + ( f ( x ) ) = L The left side limit of a function f as it approaches a is give as lim x → a − ( f ( x ) ) = L If right-side limit and left-side limit of a function are same, then a two-sided limit exists for the function. Given: A limit as lim h → 0 2 ( a + h ) 2 - 2 a 2 h Formula Used: There are multiple ways to solve limit as follows: 1. Limits by Direct Substitution – One needs to directly substitute the value to which the variable is approaching. 2. Limits by Factoring – For solving the problems involving quotient of two polynomials, one should go for the factoring method. 3. Limits by Rationalization- This technique is used for problems involving limit with square roots. Multiply and divide the expression inside the limit by the conjugate of the numerator. 4. Trigonometric limits – Apply trigonometric fundamental rule: lim x → 0 ( s i n x ) = 1 5. Limits at Infinity – Reduce the expression that contains variable in the denominator and then apply the limit rule lim x → ∞ ( 1 x ) = 1 6. Limits Involving Number e – Number e is defined as the limit: lim x → ∞ ( 1 + 1 n ) n Calculation: The given problem is lim h → 0 ( 2 ( a + h ) 2 - 2 a 2 h ) If we solve the limit by direct substitution, we will get 0 0 , thus the limit cannot be solved by direct substitution

4th Edition

ISBN: 9781319050733

Author: Rogawski

Publisher: MAC HIGHER

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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, Problem 18CRE

To determine

The limit limh→02(a + h)2 - 2a2h if it exists. If limit does not exist, then find out if one-sided limit exists.

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Chapter 2, Problem 17CRE

Chapter 2, Problem 19CRE

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Chapter 2 Solutions

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The limit lim h → 0 2 ( a + h ) 2 - 2 a 2 h if it exists. If limit does not exist, then find out if one-sided limit exists.

Solution Summary: The author explains the one-sided limit of a function, which is defined as the value that the function approaches as its argument approaches point a .

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The limit limh→02(a + h)2 - 2a2h if it exists. If limit does not exist, then find out if one-sided limit exists.

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Chapter 2 Solutions