Finding A Limit Graphically...2

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Exercises 1.2
25 - 28

20211006_201853.jpg


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
MSP64311499i660c5b7af4c000045ia67cga96dhgae


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

MSP64341499i660c5b7af4c00000i7g50gdef55gd08


26. correct

27. might exist but indeterminate
MSP222012c6a4c2829g0c38000019792ihb78id2d86


28. correct

also,
MSP267212c6a4c2829g0c3800001h1g1e82e9bh3dc7

 
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
MSP64311499i660c5b7af4c000045ia67cga96dhgae


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

MSP64341499i660c5b7af4c00000i7g50gdef55gd08


26. correct

27. might exist but indeterminate
MSP222012c6a4c2829g0c38000019792ihb78id2d86


28. correct

also,
MSP267212c6a4c2829g0c3800001h1g1e82e9bh3dc7

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?
 
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:.

upload_2021-10-7_16-53-49.png

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)
upload_2021-10-7_16-52-42.png


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
upload_2021-10-7_16-51-43.png

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

View attachment 593
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)
View attachment 592

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
View attachment 591
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.
 


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