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Evaluate the limit, if it exists.

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

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03:42

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 3

Calculating Limits Using the Limit Laws

Limits

Derivatives

Missouri State University

University of Nottingham

Idaho State University

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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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.

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