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Suppose $ f $ and $ g $ are continuous functions such that $ g(2) = 6 $ and $ \displaystyle \lim_{x \to 2} [3f(x) + f(x)g(x)] = 36 $. Find $ f(2) $.
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02:58
Daniel Jaimes
Calculus 1 / AB
Chapter 2
Limits and Derivatives
Section 5
Continuity
Limits
Derivatives
Zachariah R.
February 2, 2022
how did you get 9?
Maitha A.
September 12, 2021
how did you get the 9 ?
Oregon State University
Harvey Mudd College
Baylor 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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suppose F and G are continuous functions such that G F two is equal to six. And the limit of three times F of X plus F of X times G of X as X approaches to is 36. And here we want to find ff two. And by definition of continuity, if F and G are continuous functions, then we have the limit as X approaches to of F of X, this is equal to F F two. And the limit as X approaches to of G of X, this is equal to G F two. Now, since june of two is equal to six, then this is equal to six. And so the limit as X approaches to of three times F of X plus F of X times G fx This is equal to the limit as X approaches to of three times F of X plus, we have the limit as X approaches to of f of x times geo vex, which we can be right into three times the limit as X approaches to of F of X plus, Yes, the limit as X approaches to of F of X. This times the limit as X approaches to of G of X. Now, if we factor out the limit of F of X as X approaches to, we have limit as X approaches to of F of X this times three plus, the limit as X approaches to G fx given that this equals 36 the limits of G of X as X approaches to six, then we have 36. This is equal to the limit as X approaches to of F of X, this times three plus six. And since this is nine, we have the limit as X approaches to of F of X. This is equal to 36/9 or four. And because F of X container was then this limit of F of X as X approaches to this is equal to the value of F at two. Therefore, F of to this is equal to four.
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