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Make a conjecture about the limit by graphing the function involved with a graphing utility; then check your conjecture using L'Hôpital's rule.$$\lim _{x \rightarrow(\pi / 2)^{-}} \frac{4 \tan x}{1+\sec x}$$
4
Calculus 1 / AB
Chapter 3
TOPICS IN DIFFERENTIATION
Section 6
L Hopital's Rule; Indeterminate Forms
Functions
Limits
Derivatives
Differentiation
Continuous Functions
Applications of the Derivative
Campbell University
Harvey Mudd College
University of Michigan - Ann Arbor
University of Nottingham
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So let's go ahead and try to graph this thing. Um, and we're looking at the limit as X goes to pi over two. So that's about right there. Um, looks like the limit could be. Could be three. So my conjecture is that the limit is three. Okay. And since I've already calculated this, I can tell you that that's actually incorrect, that the limit is four. Um, but anyway, let's just stick with that conjecture. All right? So we have 4 10 divided by one plus c can't x. I noticed that if we try to evaluate this, we get negative infinity in the numerator and negative infinity in the denominator. So we actually have an indeterminant form here, and we can use low petals rule. So in the numerator, we get four times, see? Can't squared of X and in the denominator, we end up getting seek and times tangent of X. And we can simplify this a little bit. We get four times, see, can't of X divided by 10 x. And this works out when you use the definitions off Tan and C can't to be four times cosi can't x, which is equal to four. Okay, so our limit is equal to four. That's it
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