What Are the Properties of Limits in Mathematics?

[nycmathguy]

Homework Statement

Determine the limit of f(x) using properties of limits.

Relevant Equations

N/A

See attachment for question and math work.

 

[Doc Al]

You took the cube root of 8 (8^1/3) and then cubed it. Why?

 

[nycmathguy]

You took the cube root of 8 (8^1/3) and then cubed it. Why?

Let's take it from here:

(1/4)[cr{8}]

(1/4)(2)

1/2

The limit is 1/2.

Why cubed the cube root of 8 in my first attempt?

Answer: typo

 

[Doc Al]

OK, now you've got it.

 

[Delta2]

Yes the limit is 1/2. If we tweak the function and we make it so that f(x) is the same as before for all $x\neq 8$ but we set $f(8)=256$ what will the limit be?

 

[nycmathguy]

Yes the limit is 1/2. If we tweak the function and we make it so that f(x) is the same as before for all $x\neq 8$ but we set $f(8)=256$ what will the limit be?

The limit would be 256.

 

[Delta2]

The limit would be 256.

Nope it will remain 1/2. For the limit we focus what happens around the point of interest but not necessarily onto the point of interest. Around 8 the tweaked function remains the same so the limit remains the same. I just introduced an artificial discontinuity in the tweaked function by setting f(8)=whatever except 1/2.

I guess you need to be introduced to continuity and discontinuity by your textbook. If it is not done by your precalculus book, it should be done by your calculus I book.

 

[PeroK]

Nope it will remain 1/2. For the limit we focus what happens around the point of interest but not necessarily onto the point of interest. Around 8 the tweaked function remains the same so the limit remains the same. I just introduced an artificial discontinuity in the tweaked function by setting f(8)=whatever except 1/2.

I guess you need to be introduced to continuity and discontinuity by your textbook. If it is not done by your precalculus book, it should be done by your calculus I book.

What are limits doing in a pre-calculus book, one wonders?

 

[Delta2]

What are limits doing in a pre-calculus book, one wonders?

The boundaries between calculus and precalculus are fuzzy, at least that's what Ron Larson thinks lol...

 

[nycmathguy]

What are limits doing in a pre-calculus book, one wonders?

Not in a precalculus book. I made a typo. The limits are in the following book: