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Sandwich Theorem: changing inequality

  • #1
528
33

Homework Statement


Using sandwich theorem evaluvate:
$$\lim_{x\rightarrow \infty} \frac{x+7sinx}{-2x+13}$$

Homework Equations


Sandwich theorem

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:

Answers and Replies

  • #2
33,276
4,984

Homework Statement


Using sandwich theorem evaluvate:
$$\lim_{x\rightarrow \infty} \frac{x+7sinx}{-2x+13}$$

Homework Equations


Sandwich theorem

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.
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.
AdityaDev said:
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.
 

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