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Use the definitions of the hyperbolic functions find each of the following limits.(a) $ \displaystyle \lim_{x \to ^x} \tanh x $ (b) $ \displaystyle \lim_{x \to ^{-x}} \tanh x $(c) $ \displaystyle \lim_{x \to ^x} \sinh x $ (d) $ \displaystyle \lim_{x \to ^{-x}} \sinh x $(e) $ \displaystyle \lim_{x \to ^x} sech x $ (f) $ \displaystyle \lim_{x \to ^x} \coth x $(g) $ \displaystyle \lim_{x \to 0^+} \coth x $ (h) $ \displaystyle \lim_{x \to 0^-} \coth x $(i) $ \displaystyle \lim_{x \to ^{-x}} csch x $ (j) $ \displaystyle \lim_{x \to ^x} \frac {\sinh x}{e^x} $
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Calculus 1 / AB
Chapter 3
Differentiation Rules
Section 11
Hyperbolic Functions
Derivatives
Differentiation
Campbell University
Harvey Mudd College
University of Michigan - Ann Arbor
Boston College
Lectures
04:40
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.
44:57
In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.
03:08
Use the definitions of the…
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its players. So in Yuma right here. So we have limit as X approaches infinity, where the hyperbolic function of tangent is of axe. We're just gonna look at this which becomes fine over co sign, which is equal to eat the X Marchioness e to the negative X over too. All over. Eat two x plus Beat to the negative X over too. This is equal to each the x minus. Eat the negative X all over each the x plus e to the negative effects We multiply e to the negative x from the top and bottom When we get one for part B, we're doing the limit as X approaches negative infinity for the hyperbolic function of tangent of deaths When we get E to the X minus one over e to the X overeat the X plus one over heat that's were multiplying the top and bottom But eat the eggs in the end up getting negative one for part c you get the limit as X approaches infinity where the hyperbole function of sign We look a TTE on Lee the sign part and this becomes equal to eat to the ex minus e to the negative X over too, which is equal to eat. The X minus one over. Need to x over tomb which is equal to infinity minus one over infinity. Over too. Can we get infinity for a party? We get the high public function of sign to be equal to each the X over too minus e to the negative X over too. And that's the limit Approaches negative Infinity. This becomes equal to the limit as X approaches Negative infinity, for it eats the X over too minus limit US eat X approaches Negative Infinity brick eats the negative acts over too. So you end up getting 1/2 climbs zero minus negative, 1/2 times infinity. We're just equal to infinity For part e, we're gonna do limit as X approaches infinity with high public function of seek it. We're just equal to two over e to the X wants e to the negative x and using the limit properties, we get zero since it becomes too over the limit as X approaches infinity very e to the X plus e to the negative x for part f. The limit has X approaches infinity with that probably function of co tangent is equal to eat the X plus e to the negative x all over e to the X minus g to the negative Knicks. You multiply the top and bottom by E to the negative X, which is equal to one plus the limit. As X approaches infinity, I need to the negative to X all over one minus the limit. That's X approaches infinity. We eat the native to X, which is equal to one for part G. We have the limit as X approaches. They're a positive without a public function of coat engine. This is equal to guy public function of co sign over the high public function of sign. This is equal to Ito X plus each the negative X over too all over e to the X minus e to the negative bets over too, which is equal to infinity. For part h, we have the limit. That's that's approaches Negative zero fur coat engine hyperbolic function When we get co sign over sign like we didn't part G. When we get the same function e to the X plus e to the negative XO, her too over. Eat the X minus fees to the negative X over too. And then we eat the X plus e to the negative x all over eat X minus. Need to the negative X terms eats the X over eats of X, which is equal to negative infinity Ripper I We're also dealing with limits again, but this time for a co seek it. X approaches negative infinity. This is born over sign hyperbolic function just equal to to overeat the X minus seats and negative X, which becomes equal to zero. Her part j all the rates the sport. We have the limit as X approaches infinity for sign over each. The X, which is equal to eat the x minus e to the negative ETS over to each the X, which becomes equal to one minus e to the negative two x over too, which is equal to 1/2.
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