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Problem 20 Easy Difficulty

Evaluate the limit, if it exists.

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

Answer

$\frac{4}{3}$

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Video Transcript

To evaluate this limit, we first rewrite teacher the 4th power -1. Overtake U -1. Since direct substitution gives us 1 to the 4th power -1 Over one K -1 which is just 0/0. An indeterminate value notes that T to the 4th power -1 over Take U -1. This is equal to t squared minus one times t squared plus one over T -1 times T squared plus t plus one. Which is the same as T -1 times t plus one times T squared plus one over t minus one times T squared plus t plus one. Now in here we can cancel out the T -1 and reduce the expression into T plus one times T squared plus one over T squared plus t plus one. And so using this we have this limit equal to limit as T approaches one of t plus one times t squared plus one over t squared plus t plus one which is just one plus one times one squared plus one over one squared plus one plus one, which is just 4/3. And so this is the value of the limit.