Prove Limit Using Epsilon-delta Definition...1

[nycmathguy]

Calculus
Section 2.4

I don't need to know this stuff to succeed in Calculus. However, I would like to have your math work reply file away as future study notes and/or reference notes. How about working out 22?

 

[MathLover1]

22.

We need to show that for every positive ε, there is a positive δ such that if

0<|x-(-1.5)| < δ we get |(9-4x^2)/(3+2x) -6|<δ

Note first that |x-(-1.5)| =|x+1.5|


For every x other than −1.5, we have:

|(9-4x^2)/(3+2x) -6|= |(3-2x) -6| ( you get that when you simplify (9-4x^2)/(3+2x) =3-2x, then subtract -6 which is 3-2x-6=-3-2x)


then |-3-2x|=| -4|(x+1.5) | =|-4| *|(x+1.5) | =4||(x+1.5) |

We want this to be less than ε and we control, through δ, the size of |x+1.5|

if we make |x+1.5| <ε /4, then we will have

4||(x+1.5) | < 4(ε /4)

4||(x+1.5) | < ε as desired


Now we are ready to write the proof:

Given ε>0, let δ=ε/4. Observe that this δ is also positive, as required.

Now if x is chosen so that 0<|x+1.5|<δ, we will have:

|(9-4x^2)/(3+2x) -6| =4||(x+1.5) |

4||(x+1.5) | <4δ

then 4δ=ε

that is:

if 0< |(x+1.5) |< δ, then |(9-4x^2)/(3+2x) -6| <ε

so, by the definition of lim,

 

[nycmathguy]

22.

View attachment 3035

We need to show that for every positive ε, there is a positive δ such that if

0<|x-(-1.5)| < δ we get |(9-4x^2)/(3+2x) -6|<δ

Note first that |x-(-1.5)| =|x+1.5|


For every x other than −1.5, we have:

|(9-4x^2)/(3+2x) -6|= |(3-2x) -6| ( you get that when you simplify (9-4x^2)/(3+2x) =3-2x, then subtract -6 which is 3-2x-6=-3-2x)


then |-3-2x|=| -4|(x+1.5) | =|-4| *|(x+1.5) | =4||(x+1.5) |

We want this to be less than ε and we control, through δ, the size of |x+1.5|

if we make |x+1.5| <ε /4, then we will have

4||(x+1.5) | < 4(ε /4)

4||(x+1.5) | < ε as desired


Now we are ready to write the proof:

Given ε>0, let δ=ε/4. Observe that this δ is also positive, as required.

Now if x is chosen so that 0<|x+1.5|<δ, we will have:

|(9-4x^2)/(3+2x) -6| =4||(x+1.5) |

4||(x+1.5) | <4δ

then 4δ=ε

that is:

if 0< |(x+1.5) |< δ, then |(9-4x^2)/(3+2x) -6| <ε

so, by the definition of lim,View attachment 3036

Wish I could do this just like you. Why am I struggling with this stuff? Thank God it's not required for me to learn calculus.

See link below for the last one today.

Use Graph to Find Delta...2