Page 211 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes
Example: Let f(x) be a function such that |f(x)| M, for any x 0. The Sandwich Theorem
implies

Indeed, we have
|x f(x)| M|x|
which implies
–M |x| x f(x) M|x|

for any x 0. Since

then the Sandwich Theorem implies

Example: Use the Sandwich Theorem to prove that

for any a > 0.
Solution:
For any x > 0, we have

Hence

because for any x > 0. In particular, we have

Since

the Sandwich Theorem implies

or equivalently

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