The limit of function that is squeezed between two other functions, each function will have same limit at given value of x. If two functions squeeze together at a particular point, then any function trapped between them will get squeezed to that same point.

1) If two functions sandwich together at a particular point, then any function trapped between them will get squeezed to that same point.

2) The squeeze or sandwich theorem deals with the limit values rather instead of function values.

3) Squeeze theorem or sandwich theorem is also called Pinch theorem.

If h(x) $\leq$ f(x) $\leq$ g(x), for all x in open interval containing 'c' except possibly at 'c' itself, and if

$\lim_{x->c} h(x) = L = \lim_{x->c} g(x)$ then

$\lim_{x->c}f(x)$ exists and is equal to 'L'.

$-1 \leq sin(\frac{1}{x})\leq 1$ for all x

So, $-|x| \leq sin(\frac{1}{x})\leq |x|$ for all x

We know that

$\lim_{x->0}|x|$ = 0 and $\lim_{x->0}-|x|$ = 0

So according to squeeze theorem $\lim_{x->0} x * sin(\frac{1}{x})$ = 0

2) Use sandwich theorem to find $\lim_{x->c}f(x)$, c = 0 , $4 - x^{2}\leq f(x)\leq 4 + x^{2}$

Now we will use squeeze theorem to find lim f(x) as x approaches to 0

h(x) $\leq$ f(x) $\leq$ g(x)

$\lim_{x->0}4 - x^{2}$ = 4 and $\lim_{x->0}4 + x^{2}$ = 4

$4 \leq \lim_{x->0}f(x)\leq 4 $

So, $\lim_{x->0}f(x)$ = 4

3) Evaluate $\lim_{x->0} x * cos(\frac{1}{x})$

$-1 \leq cos(\frac{1}{x})\leq 1$ for all x

So, $-|x| \leq cos(\frac{1}{x})\leq |x|$ for all x

We know that

$\lim_{x->0}|x|$ = 0 and $\lim_{x->0}-|x|$ = 0

So according to squeeze theorem $\lim_{x->0} x * cos(\frac{1}{x})$ = 0

We at **ask-math **believe that educational material should be free for everyone. Please use the content of this website for in-depth understanding of the concepts. Additionally, we have created and posted videos on our youtube.

**We also offer One to One / Group Tutoring sessions / Homework help for Mathematics from Grade 4th to 12th for algebra, geometry, trigonometry, pre-calculus, and calculus for US, UK, Europe, South east Asia and UAE students.**

**Affiliations with Schools & Educational institutions are also welcome.**

Please reach out to us on [email protected] / Whatsapp +919998367796 / Skype id: anitagovilkar.abhijit

We will be happy to post videos as per your requirements also. Do write to us.

Russia-Ukraine crisis update - 3rd Mar 2022

The UN General assembly voted at an emergency session to demand an immediate halt to Moscow's attack on Ukraine and withdrawal of Russian troops.

From Calculus squeeze theorem to Home

GMAT

GRE

1st Grade

2nd Grade

3rd Grade

4th Grade

5th Grade

6th Grade

7th grade math

8th grade math

9th grade math

10th grade math

11th grade math

12th grade math

Precalculus

Worksheets

Chapter wise Test

MCQ's

Math Dictionary

Graph Dictionary

Multiplicative tables

Math Teasers

NTSE

Chinese Numbers

CBSE Sample Papers