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Evaluate the limit and justify each step by indicating the appropriate Limit Law(s).
$ \displaystyle \lim_{t \to -2}\frac{t^4 - 2}{2t^2 - 3t + 2} $
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05:08
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 3
Calculating Limits Using the Limit Laws
Limits
Derivatives
Missouri State University
Campbell University
Oregon State University
University of Nottingham
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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Evaluate the limit.$$<…
in this problem we are asked to evaluate this limit limit T approaches -2 of T. The 4th -2 over to t squared -3 T Plus two. All right. So first of all, let's just check if we evaluate the denominator With T at -2. That's two times minus two squared minus three times minus two plus two. That is what we'll minus two squared is four times two is eight. Man is a modest. That's plus two times 3 is six plus two. So that's eight and eight. That's 16. Okay, so this is not zero. All right. So we don't have an issue with dividing by zero and having a vertical assam. Tote at minus two. So therefore the first limit law is at the limit of F of X. This little needle here half of X or G fx is limit of Absolutely limit of half of x times one over the limit of G f x. So that means that we now have a limit As T approaches -2 of Do you? The 4th 1 is two over The limit as T approaches -2 of to t squared minus three T plus two. Okay. Yeah, The next one is a polynomial limit law here. Right? Or the limit of a X squared plus bx. We'll see is is what is the constant? They can come out in front limit X squared plus B times the limit of X. Since he's a constant can come out front was a limit of C which is just see by the way. Okay, so that means I now have The limit as T approaches -2 of T to the 4th minus two over. The limit is two times the limit as T approaches minus two of t squared -3 times the limit as t approaches -2 of T plus two on the end. Okay, so evaluating each one of these limits, well, the limit of this polynomial is just Just going to be this number. Interview that ponytails, that's -2 to the fourth, homelessness to over two times A -2. Is that polonium? It's -2 squared -3 times -2. That's too. And so what do I get? Well minus to the fourth, that's 16 minus two, That's 14 over. We just calculated this to be 16 And so this is 7/8, isn't it? In reduced form.
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