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Evaluate the limit, if it exists.

$ \displaystyle \lim_{t \to 1}\frac{t^4 - 1}{t^3 - 1} $

$\frac{4}{3}$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

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this is problem number twenty of this tour Calculus eighth edition, section two point three. You know the way to the limit. If it exists, the limit is T approaches one of T to the fourth minus one divided by T to the third power minus one. We will begin qui usually Dubai simplifying this limit and we're going to need it. A few tools are some references for these. For simple find these functions we have the difference of square function. This is a rule. The difference of squares Commie written, rewritten as a scream. SB scored community rewritten as a plus B a minus bean and a difference of cubes can be written as a Q minus. B Cubed equals payments being Time's a squared place, Baby Cosby's quieted down So we're gonna use these in the numerator. We have a difference of squares. We have t squared squared minus one squared. So this fax it factors into t scored with this one Auntie squared plus one. According to this April speaks my name minus b and then the denominator. We have t cubed minus one or tea is being cute and one is being cubed. So we simplified to T minus one and then a squared or just T squared, plus eight times being team plus B Squared one, which is one squared or one. We should also show that this T Square minus one is also a difference of squares, and it can also be simplified. Two t plus one T minus one. After having done there, we see that we can simplify and cancel these team, and it's one and released over with the limit is t approaches one ah t squared, plus one temp's t plus one, right, right, the quantity t squared plus t plus one. And if we evaluate this and tea is equal to one, we get one plus one time is that one plus one divided by one plus one plus one and then the numerator re of two times two or four. And in the denominator we have one plus one plus one or just three. And for over the their final answer

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