82. Use the definitions of the hyperbolic functions to find ea...

Question

82. Use the definitions of the hyperbolic functions to find each of the following limits.

i. lim(x→-∞) csch x

Accepted Answer

Welcome back, everyone. Evaluate the limit as X approaches infinity of sash of 2 X. A says 0, B1, C infinity, and D-1. Before we begin, let's analyze sash of two eggs. And what we're going to do is simply express it in terms of cash. It is basically 1 divided by cache of 2 X. Let's remember the definition of cash. Cash of 2 X can be expressed in terms of E. That's e to the power of 2 X plus e to the power of -2 X divided by 2. So now what is se of 2 X in terms of exponentials? We want to take the reciprocal of cash, so that would be 2, our denominator divided by the numerator. So 2 divided by e to the power of 2 X plus e to the power of negative 2X. Meaning we can now evaluate the limit as x approaches infinity. Of 2 divided by e to the power of 2 X plus e to the power of -2 X. Now, let's analyze what's happening. In the numerator, we have a constant, which is 2. Now eats the power of two axes, X approaches infinity is going to be eats the power of infinity, which approaches infinity, and we are adding eats the power of -2 X as X approaches infinity. E to the power of negative infinity. Well, that's going to approach 0, right? So now we are basically dividing 2 by an infinitely large number, which means that the limit is equal to 0 because we have a constant divided by infinity. So the answer to this problem corresponds to the answer choice a 0. Thank you for watching.

82. Use the definitions of the hyperbolic functions to find each of the following limits.i. lim(x→-∞) csch x

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82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x