Finding A Limit Graphically...2

Exercises 1.2
25 - 28



25)

The limit from the left does not equal the limit from the right. Thus, no limit.

26)

The limit from the left is negative infinity. The limit from the right is positive infinity. The limit from the left does not equal the limit from the right. So, no limit.

27)

This is an oscillating function. The limit does not exist.

28)

The limit from the left is positive infinity.
The limit from the right is negative infinity.
The limit from the left does not equal the limit from the right. So, the limit does not exist.

You say?

 

25)
there are holes in (2,1) and (2,-1)
as x comes closer and closer to 2 from the left side, limit is getting closer and closer to -1 (but never gets equal to -1) even we write that

as x comes closer and closer to 2 from the right side, limit is getting closer and closer to 1



26. correct

27. might exist but indeterminate


28. correct

also,

 

For 27 you said the limit "might exist but indeterminate." How can we know for sure?
Does it involve advanced limit studies?

For 25, can you show me how to find the limit algebraically? What if I let x = 2?

|2 - 2|/(2 - 2) = |0|/0 = 0/0 = indeterminate. Yes?

 

yes, or you can say undefined

 

Do you mean to show 27 algebraically involves further limit studies?
We can say that 25 is undefined? How does
L' Hopital's Rule apply when we get 0/0?

 

just for 27

25. as x comes closer and closer to 2 (but never get 2) from the right side, limit is getting closer and closer to 1

 

In terms of limits, all oscillating functions are undefined. Yes?

 

some oscillating functions have no limit , some do have a limit

Some trigonometric functions start to oscillate so radically between two y-values as x approaches a specific value the limit is undefined. The graph of f(x) = cos(1/x) is shown in this second graph appearing here:.


We can see from our second graph that as x approaches 0 from either side the graph becomes like a compressed accordion shape that is oscillating in an extreme fashion, which is an undefined limit in calculus! This is called an infinite oscillator.

or, f(x)=sin(1/x)


This function doesn't have a limit as x→0 since it just oscillates more and more wildly between -1 and 1.
so, this function undefined limit

or, f(x) = 1/x^2

This function exists for all nonzero values of x, and doesn't have a limit as x→0

 

Thank you so much. I'm so glad too have met you. I have learned do much math thanks to you. This is just the beginning of a long, mathematical journey.