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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) $.

If $f$ and $g$ are continuous and $g(2)=6,$ then $\lim _{x \rightarrow 2}[3 f(x)+f(x) g(x)]=36 \Rightarrow$\[3 \lim _{x \rightarrow 2} f(x)+\lim _{x \rightarrow 2} f(x) \cdot \lim _{x \rightarrow 2} g(x)=36 \Rightarrow 3 f(2)+f(2) \cdot 6=36 \Rightarrow 9 f(2)=36 \Rightarrow f(2)=4\]

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 5

Continuity

Limits

Derivatives

Maitha A.

September 12, 2021

how did you get the 9 ?

Oregon State University

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

Lectures

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Suppose that $\lim _{(x, y…

This is probably where forty seven of the Stuart Calculus eighth edition section two point five suppose f n g are continuous functions such that G two is equal to six and the limit and six approaches to of the quantity three plus plus F times. Jean is equal to thirty six. Find effort, too. So what we're going to do, we're going to work with this limit first and approach it in a way that we don't know what the functions are like f in gene. But we use what is given to us to determine what this should be equal to. Now. The important information is that F N g are continuous function from our Long's about continuous functions, but continues functions can be more complied and added together create another continuous function. So what we can be right here is that because this is a continuous function, three F plus f times G thirty six is also equal to this function. You would say we call this function in the meal. It's called his new function HMX. We know that's also continuous that the limit of this function is X approaches to will be equal to the function itself. H evaluated it, Tio. And we know that from our definition of continuity. So I'm going to go ahead and you really read this since each of two equals thirty six and ages dysfunction, ah, age too is equal to three times effort too, Agnes, from to time to CI of two. And as we said, this's kind of equal to the limit, which was given his thirty six keep in mind. The GI ft was also given equal to six. So our only unknown is half of two. Go ahead and souls or have to get nine times of two. It's equal to thirty six in the only way that this is true is, if ever, too two for so he's in a lot of continuity. Three. But we get our final result that have too much equal to four

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