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 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 approaches 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 the same, then a two-sided limit exists for the function. Given: A limit as lim x → 1 + ( 1 x − 1 − 1 x 2 − 1 ) 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 x → 1 + ( 1 x − 1 − 1 x 2 − 1 ) Let us first simplify 1 x − 1 − 1 x 2 − 1 by taking 1 x − 1 as common, we get 1 x − 1 − 1 x 2 − 1 = 1 x − 1 ( 1 − 1 x 2 − 1 1 x − 1 ) Now, let us find the limit of lim x → 1 + ( 1 x − 1 ( 1 − 1 x 2 − 1 1 x − 1 ) ) lim x → 1 + ( 1 x − 1 ( 1 − 1 x 2 − 1 1 x − 1 ) ) = lim x → 1 + ( 1 x − 1 ) ⋅ lim x → 1 + ( 1 − 1 x 2 − 1 1 x − 1 ) First find the value of limit lim x → 1 + ( 1 x − 1 ) For x approaching 1, such that x > 1 , ⇒ x − 1 > �

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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Question

Chapter 2, Problem 43CRE

To determine

The limit limx→1+(1x − 1−1x 2 − 1) if the limit exists. If limit does not exist, then find out if one sided limit exists.

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Chapter 2, Problem 42CRE

Chapter 2, Problem 44CRE

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Chapter 2 Solutions

CALCULUS (CLOTH)

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The limit lim x → 1 + ( 1 x − 1 − 1 x 2 − 1 ) if the limit exists. If limit does not exist, then find out if one sided limit exists.

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

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The limit limx→1+(1x − 1−1x 2 − 1) if the limit exists. If limit does not exist, then find out if one sided limit exists.

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Chapter 2 Solutions