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CalculusQ&A LibraryI.Consider the function f(x)4/x2.a) Find the domain of f(x).b) Find f'() using the limit definition of the derivative.c) Where is f(x) differentiable?2. Find all horizontal and vertical asymptotes of the following functions. Be sure to label the asymptotescorrectlya) ()33z-1 2(b) g(x) = tan-1 (2)c) h(x)-T2-92-112-3--1213. Evaluate the following limits.2-2x-3b) lim,ーナー0。6a3+2+16c) lim,--0 f(x) , where f(x)-(2x = 00sin(x)V2+1+3cd)limx-W--5x. Find an equation of the tangent line to the function f(x) = Va2+16 at the point (3,5). (You NQuestion

Asked Feb 1, 2019

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Can you help me answer question 2?

Step 1

Vertical assymptotes are the lines parallel to y-axis, i.e thier equation will be of the for x = a. This would happen when at f(a) the function would tend towards infinity.

For horizontal assymptotes, these are the lines of the form y = a, where a is the limiting value of thee function as x changes .

Step 2

Part 1.

Since this is a continuous function with no break points, we will check at infinity and negative infinity. We find that the function rises infinitely at x=infinity. But at negative infinity we see that the exponent dies and we get a limiting value. Hence we find the assypmtote :

y = -2

IT DOES NOT HAVE A VERTICAL ASSYMPTOTE. (or you can say it is x = infinity)

Step 3

Part 2

Inverse tangent is the inverse of a peiodic function. Hence it has many assypmtotes because it exists periodically over many intervals. For the most common interval, i.e between plus and minus pi/2 we see that the assymptotes are as follows.

Be carefull that it is a periodic functio...

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