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Sandwich Theorem: changing inequality

  1. Dec 21, 2014 #1
    1. The problem statement, all variables and given/known data
    Using sandwich theorem evaluvate:
    $$\lim_{x\rightarrow \infty} \frac{x+7sinx}{-2x+13}$$
    2. Relevant equations
    Sandwich theorem
    3. The attempt at a solution

    ##-7 \leqslant 7sinx \leqslant 7##
    ##x-7 \leqslant x+7sinx \leqslant x+7##

    Now my doubt: I want to divide the expression by ##-2x+13##. But does this change the inequality? I don't know if it is positive or negative.
    If I divide, I will get the expression in the question.
    (hint given in my textbook: Inequality changes that is they assumed ##-2x+13## is negative.

    Let me not change the inequality (I will get the answer but I need to know if what I am doing makes sense)

    ## \frac{x-7}{-2x+13} \leqslant \frac{x+7sinx}{-2x+13} \leqslant \frac{x+7}{-2x+13} ##

    ##\lim_{x\rightarrow \infty} \frac{x-7}{-2x+13} = \lim_{x\rightarrow \infty} \frac{1- \frac {7}{x} }{-2+\frac{13}{x}} = -\frac{1}{2}##

    Similarly the right side limit is also -1/2(same method). Hence limit of middle term is also -1/2. Answer is correct.
    Last edited: Dec 21, 2014
  2. jcsd
  3. Dec 21, 2014 #2


    Staff: Mentor

    Since your limit is as x gets very large, -2x + 13 can be assumed to be negative. Dividing by -2x + 13 will then change the direction of the inequalities.
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