Calculus

Hi!

My question is:

Given that f is a function defined by f(x) = (2x - 2) / (x^2 +x - 2)

a) For what values of x is f(x) discontinuous?
b) At each point of discontinuity found in part a, determine whether f(x) has a limit and, if so give the value of the limit.
c) write the equation for each vertical and each horizontal asymptote for f. Justify your answer.
d) A rational function g(x) = (a) / (b + x) is such that g(x)=f(x) whenever f is defined. Find the values of a and b.

Ok so I figured out the answers to a and b. A is "discontinuous at x = -2 and x = 1. and b is "as the limit approches -2, it does not exist and is nonremovable and as the limit approches 1, the limit is 2/3 and is removable". I'm not sure how to do part c and d though. Hopefully someone can help me!!

Thanks!! :)

  1. 👍 0
  2. 👎 0
  3. 👁 1,195
  1. You probably factored the function properly and got
    f(x) = (2x - 2) / (x^2 +x - 2)
    = 2(x-1)/[(x-1)(x+2)]
    = 2/(x+2) , x not equal to 1

    from f(x) = = 2(x-1)/[(x-1)(x+2)]
    you are right to say that it is discontinuous at x = 1 and -2

    notice when x = 1, f(1) = 0/0 which is indeterminate and
    Limit = 2(x-1)/[(x-1)(x+2)] as x -->1
    = 2/3
    but when x=-2 , f(-2) = 2/0 which is undefined.

    So we have a "hole" at (1,2/3) and an asymptote at x = -2

    for the horizontal asymptote I let x --> ∞ in the original function
    I see "large"/"really large" which goes to zero as x gets bigger.
    so y = 0 is the horizontal asymptote.

    for your last part, notice that our reduced function
    f(x) = 2/(x+2) has the form a/(b+x), so
    a = 2
    b = 2

    I will leave it up to you to fit all those parts in the proper question/answers.

    1. 👍 0
    2. 👎 0
  2. (a) and (b) correct.
    for (c), the vertical asymptotes are found when the denominator becomes zero. The vertical asymptotes are vertical lines passing through these singular values of x, in the form x=?.
    The horizontal asymptote can be found (in this case) by finding the value of f(x) as x-> +∞ and x->-∞.
    It will be in the form y=?.

    (d) First look at f(x) as
    f(x) = 2(x-1)/((x-1)(x+2))
    When f(x) is defined, x≠1 and x≠-2. When x≠1, what can you say about the common factors (x-1) in the numerator and denominator?

    1. 👍 0
    2. 👎 0
  3. Oh my goodness!!! Thank you so so much! That makes perfect sense! Thank you!!! :)

    1. 👍 0
    2. 👎 0
  4. MathMate thank you so much as well!! The (x-1)'s cancel out which gives me the equation 2/(x+2), which like Reiny said, is in the form I need it in!

    Seriously I cannot thank you enough for your help! I totally understand how to solve this problem now! :)

    1. 👍 0
    2. 👎 0
  5. I beg to differ with MathMate.

    Since the limit exists at x=1, there is no vertical asymptote at x=1, only at x=-2

    The statement, "the vertical asymptotes are found when the denominator becomes zero" should be clarified to say
    "the vertical asymptotes are found when the denominator becomes zero but the numerator is non-zero"

    1. 👍 0
    2. 👎 0
  6. Reiny, thank you for the correction. That was an oversight.

    1. 👍 0
    2. 👎 0

Respond to this Question

First Name

Your Response

Similar Questions

  1. Calculus (Continuity and Differentiability)

    Okay. So I am given a graph of a derivative. From what I can gather, it looks like the function might be abs(x-2)-4. (I was not given an explicit function for g', just its graph.) The question then goes on to ask me: Is it

    asked by Mishaka on November 12, 2011
  2. math

    what is the factorial of a negative number? The factorial function has singularities at the negative integers. You can see this as follows. For integers we define: (n+1)! = (n+1)n! and we put 0! = 1 So, from 0! you can compute 1!

    asked by chris on July 10, 2007
  3. Math - PreCalc (12th Grade)

    The function f(x) = 2x + 1 is defined over the interval [2, 5]. If the interval is divided into n equal parts, what is the value of the function at the right endpoint of the kth rectangle? A) 2+3k/n B) 4+3k/n C) 4+6k/n D) 5+6k/n

    asked by Shawna on March 21, 2014
  4. Calc

    The function f is defined by f(x)= (25-x^2)^(1/2) for -5 less than or = x less than or = 5 A) find f'(x) B) write an equation for the tangent to the graph of f at x=3 C) let g be the function defined by g(x)= (f(x) for -5

    asked by Aparna on March 19, 2013
  5. algebra

    a function is defined by the equation y=8x-3. If the domain is 2≤x≤4, find the minimum value in the range of the function.

    asked by Carla on July 27, 2012
  1. Calculus - Functions?

    #1. A cubic polynomial function f is defined by f(x) = 4x^3 +ax^2 + bx + k where a, b and k are constants. The function f has a local minimum at x = -1, and the graph of f has a point of inflection at x= -2 a.) Find the values of

    asked by Amy on February 21, 2011
  2. Check my CALCULUS work, please! :)

    Question 1. lim h->0(sqrt 49+h-7)/h = 14 1/14*** 0 7 -1/7 Question 2. lim x->infinity(12+x-3x^2)/(x^2-4)= -3*** -2 0 2 3 Question 3. lim x->infinity (5x^3+x^7)/(e^x)= infinity*** 0 -1 3 Question 4. Given that: x 6.8 6.9 6.99 7.01

    asked by Samantha on November 11, 2013
  3. Calculus

    Answer the following questions for the function f(x) = sin^2(x/3) defined on the interval [ -9.424778, 2.356194]. Rememer that you can enter pi for \pi as part of your answer. a.) f(x) is concave down on the interval . b.) A

    asked by Anonymous on November 2, 2011
  4. Math

    If the graph of the function h defined by h(x)= -4x2-1 If translated vertically downward by 3 units, it becomes the graph of function f. Find the expression for f(x)

    asked by Leo on April 9, 2020
  5. math grade 11

    A quadratic function is defined by f(x)= 3x^2+4x-2. A linear function is defined by g(x)= mx-5. what values(s) of the slope of the line would make it a tangent to the parabola.

    asked by priya on November 3, 2010

You can view more similar questions or ask a new question.