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Applying Limits to Find the Vertical Asymptote

  1. Nov 25, 2008 #1
    Sorry to be a bother at this time around. I have a question though about a wierd looking function : e^(6-2x) + 5. (The 5 is outside of the parenthesis, so it is a coefficient, not an exponent). I am interested in seeing how to use limits to construct its graph. In particular, I want to find out what it looks like to the left of 3. (Its vertical asymptote in short)


    y - intercepts

    y = e^(6) + 5

    (0, e^(6) + 5)


    x - intercepts

    0 = e^(6) / e^(2x) + 5

    -5e^(2x) = e^(6)

    Since natural logging it would give an undefined answer there is no x-intercept. At x --> infinity (e^(6)/ e^(2x) + 5) tends to 5. Then the horizontal asymptote is 5.

    Applying the derivative test, e^(6-2x) + 5 becomes (-2)e^(6-2x) + 5. Its only critical point or when f'(x) = 0 is at x = 3. Evaluating at x=2 and x=4 it appears to have negative values indicating that the function is decreasing. When x = 3 is put back into the original f(x) it gives 5, indicating that the asymptote starts at 3.

    Since the horizontal asymptote starts at x = 3. At x --> 3- (e^(6-2x) + 5) = + infinity. The function from the left of 3 is a monotic function because like stated before, the MVT says that at f'(c) = 0 and the only critical point is at x = 3 so it is safe to assume that it has either an increasing or decreasing pattern.

    I know that at x = 3 it has a HA = 5, but how can you find the exact point where the function begins to to "blow up" vertically?
     
  2. jcsd
  3. Nov 25, 2008 #2

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    NO. The derivative of "5" is 0.

    Again, no. Even using your "derivative", if x= 3, 6- 2x= 0 but -3e0+ 5= 3, not 0. f'= -2e6- 2x is never 0.

    you can't- it doesn't. There is NO vertical asymptote.
     
  4. Nov 25, 2008 #3
    Hi Halls,

    Thanks for catching the derivative error I made. I did put the function in an online graphing calculator and did see a vertical asymptote.

    http://www.coolmath.com/graphit/
     
  5. Nov 25, 2008 #4

    Mark44

    Staff: Mentor

    That's not a vertical asymptote. A vertical asymptote is a line, x = a, where a is some finite number. On either side of this line the graph of a function becomes unbounded in some way (i.e., approaches infinity or negative infinity).

    Your exponential function grows very large as x becomes negative. What you probably saw was the graph getting very steep for some negative value of x. In a similar way, the graph of, say, f(x) = x^3 gets pretty steep for large values of x, but it too does not have a vertical asymptote.
     
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