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Use the graphs of $ f $ and $ g $ and their tangent lines at $ (2, 0) $ to find $ \displaystyle \lim_{x\to 2} \frac{f(x)}{g(x)} $.

$\lim _{x \rightarrow 2} \frac{f^{\prime}(x)}{{g}^{\prime}(z)}$

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

So looking at the two graphs, um what we can see is that mhm the tangent of wine S is going to be, so if we have F. Of X equaling um this is gonna be our tangent line for the f graph 1.5 times X minus two. And then are the tangent line of our G of X craft is going to be Negative X -2. I mean these are both at the points to zero. So then what based on this? If we have F. Of two, we have G. Of two, we see that uh based on this. So this is the tangent line of F and the tangent line of G. So now we need to um differentiate these because we get 0/0. So now we need to use low tiles rule. And if we do f prime of two over g prime of two, we end up getting 1.5 over negative one. So our final answer for the limit is going to end up being -1.5. That will be our final answer. And that's the limit as X approaches to of F of X over G. Fx through using reptiles rule.

California Baptist University