Finding limit of an oscillating function
About this video
The video looks at f(x) = 1.5 cos(1/x^2) and asks whether the limit as x approaches 0 can be determined. As x gets close to 0, the input to cosine blows up, so f oscillates faster and faster between 1.5 and -1.5. Because it never settles on one value from either side, the limit does not exist.
You'll be able to
- explain why a limit needs the function to approach the same single value from both sides.
- explain why a function that oscillates faster and faster near a point has no limit there.
Key words
- oscillate
- To keep swinging back and forth between values.
- converge
- To settle closer and closer to one particular value.
Transcript
- [Instructor] We are told the graph of f of x is equal to 1.5 times cosine of one over x squared, and that is shown right over here. Can the limit as x approaches zero of f of x be determined? Pause this video. Have a go at that. Alright, so there's a couple of ways to approach this.
One way is to just realize, okay, we're trying to think about what does the function approach as x approaches zero. And of course, in order for that limit to be defined, it has to approach the same value for f of x as we approach from the left, as we approach from the right.
Now, the problem with this function is the closer we get to x equaling zero, this expression right over here blows up. When x becomes super small, one over x becomes super huge, and this function just oscillates faster and faster between positive 1.5 and negative 1.5.
So it's not coming to any value, it's not converging to any value. So no, because it is oscillating faster and faster as x approaches zero from either side.
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