1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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

    Mark44

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Sandwich Theorem: changing inequality
  1. Liouville's Theorem (Replies: 8)

Loading...