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About Paul Klarreich
Expertise
All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

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I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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(See above.)

 
   

You are here:  Experts > Teens > Homework/Study Tips > Calculus > Limit proofs - delta-epsilon

Calculus - Limit proofs - delta-epsilon


Expert: Paul Klarreich - 9/29/2009

Question
plz solve this for me

Using the formal epsilon − delta definition of derivative, show that
lim xapprochaes 0 x^2sin(x)=0

Answer
Questioner: waqas
Country: Australia
Category: Calculus
Private: No
Subject: Epsilon Delta problems
Question: plz solve this for me

Using the formal epsilon − delta definition of derivative, show that
lim xapprochaes 0 x^2sin(x)=0
..................................................
Hi, Wagas,

You are not using the vocabulary correctly.  You should either write:

Using the formal epsilon − delta definition of limit

or

Using the formal definition of derivative as a limit.

Misusing (and therefore misunderstanding) the vocabulary is a good way to fail mathematics.
............................................
I shall assume you mean:

Using the formal epsilon − delta definition of limit

show that lim [x -> 0] x^2sin(x)=0

You want to show that for any  e > 0 [e is epsilon, d is delta]

you can find  d > 0 such that whenever  |x - 0| < d,

| x^2sin(x) - 0 | < e

Start with

| x^2sin(x) - 0 | = | x^2sin(x) |

| x^2sin(x) | = | x^2 | | sin(x) |

| x^2 | | sin(x) | <= | x^2 | (1) = | x^2 | = x^2, since x^2 is never negative.

Now we want  x^2 < e, so take  |x| < sqrt(e).

That is your delta.

You can now write the proof:

| x^2sin(x) - 0 | < | x^2sin(x) | =

| x^2||sin(x) | <= |x^2|

= |x|^2 < | sqrt(e) |^2 = e

Done


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