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If $ \displaystyle \lim_{x \to 1}\frac{f(x) - 8}{x - 1} = 10 $, find $ \displaystyle \lim_{x \to 1}f(x) $.
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Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 3
Calculating Limits Using the Limit Laws
Limits
Derivatives
Campbell University
Baylor University
Idaho State University
Boston College
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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This is problem number 59 of this to calculus edition section 2.3. If the limit as x, approaches 1 of the function f of x minus 8 over x, minus 1 is equal to 10 point find the limit as x approaches 1 of the function f. So, since we're given this definitive statement that this limit is equal to 10, we'll use our limit laws to rewrite this limit. Instead, as the limit as x, approaches 1 have f minus the limit is f x, approaches 1 of a divided by the limit as x, riches 1 of the quantity x minus 1 equals to 10 point. So we've used the laws that allow us to separate this quotient within the limit into the division of 2 limits and then the difference of limits in the top and the difference in set of a limit to the difference of limits at the top and notice that This term here is the term we want to solve, for so, let's go ahead, and do that multiply both sides limit is x, plus 1 of f minus limit is x, plus is 1 of 8 is equal to 10 times. This limit as x, approaches 1 of x minus 1. We can move the other limit to the right hand side. So, on the left hand, side we have limit as x, which is 1 of f equals 10 times the limit and which is 1 of the quantity x. Minus 1 plus e limit as x approaches 1 of 8. Now, as x, proches 1, this quantity x, minus 1, is equal to 1 minus 1 or 0 and 10 times 0 is 0. So what we end up getting is that the limit, as x, pushes 1 of f is equal to 0, plus this limit the limit of just 188 being in a constant value, therefore telling what is equal to 8- and this is our final answer as to what The limit is toast, 1 of f is.
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