Estimate Limits Numerically

[nycmathguy]

Exercises 1.2
6 & 8

Question 6

The purpose of this table is to see what happens as we get closer and closer to 3. I say x cannot be the number in the middle of the table or 3. I will not evaluate this function at the given x-values here. I just want to know if my reasoning is correct.

Note: The graph tells a different story. I think x tends to -3 not 3. You say?

Question 8

The purpose of this table is to see what happens as we get closer and closer to 3. I say x cannot be the number in the middle of the table or 3. I will not evaluate this function at the given x-values here. I just want to know if my reasoning is correct.

Note: The graph tells a different story. Does x really tend to 3?

 

[MathLover1]

reasoning is correct

6.
if you calculate this way, as x->3 limit will be undefined
table
x ........... 2.9 ........... 2.99 ........... 2.999 ........... 3 ........... 3.001 ........... 3.01 ........... 3.1
f(x) ........ 0.17 ........ 0.17 ........ 0.167 ........ (undefined) ........ 0.167 ........ 0.167 ........ 0.16

but, if you simplify function you got y=1/(x+3),
table
x ........... 2.9 ... ........ 2.99 ........... 2.999 ........... 3 ........... 3.001 ........... 3.01 ........... 3.1
f(x) ........ 0.17 ........ 0.167........ 0.16669........ (1/6) ........ 0.16666 ........ 0.166 ........ 0.164

then as x->3 limit will be 1/6

 

[MathLover1]

8.

so

when you do the table, you see as x->3 that f(x)=-1/16

 

[nycmathguy]

8.

Very cool. In the coming days, we can learn to take the limit algebraically. Let us go section by section and chapter by chapter.

so

when you do the table, you see as x->3 that f(x)=-1/16

 

[nycmathguy]

reasoning is correct

6.
if you calculate this way, as x->3 limit will be undefined
table
x ........... 2.9 ........... 2.99 ........... 2.999 ........... 3 ........... 3.001 ........... 3.01 ........... 3.1
f(x) ........ 0.17 ........ 0.17 ........ 0.167 ........ (undefined) ........ 0.167 ........ 0.167 ........ 0.16

but, if you simplify function you got y=1/(x+3),
table
x ........... 2.9 ... ........ 2.99 ........... 2.999 ........... 3 ........... 3.001 ........... 3.01 ........... 3.1
f(x) ........ 0.17 ........ 0.167........ 0.16669........ (1/6) ........ 0.16666 ........ 0.166 ........ 0.164

then as x->3 limit will be 1/6

Using the table is very tedious. This exercise is basically to show students how to find the limit as x tends to a certain value without actually getting there.

 

[MathLover1]

Take a look at your work. I am to use the values of x given in the table to evaluate f(x).

You said that f(x) = -1/4(x + 1). There's a minus in front of the algebraic fraction. How can the limit be

When it has a minus sign at the top it means that you must find the limit when the desired x comes from the left, meaning that you must find which number the limit goes as x becomes more closer to 3 from the left.

 

[nycmathguy]

When it has a minus sign at the top it means that you must find the limit when the desired x comes from the left, meaning that you must find which number the limit goes as x becomes more closer to 3 from the left.

Yes, I figured that out by reading your reply twice.