Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x i...

Question

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So, first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. For a function F of X where x to the power of 5 is less than or equal to F of X, which is less than or equal to x the power of 3, for X in parenthesis negative infinity 12 and 2012. And X to the power of 3 is less than or equal to F of X, which is less than or equal to x to the power of 5, for square bracket -102, and square 1 infinity, identify the points C where the limit as X approaches C of F of X is known. What are the limits at these points? Awesome. So it appears for this particular problem, we're asked to solve for two separate answers. We're asked to identify the points C, where this limit of F of X is known, and then we're for a second answer, or I should say answers, is we're trying to figure out the limits at these points. So we're trying to find, or rather identify the points C where this limit of FFX is known, and then we're trying to figure out the limits at these points of C. Fantastic. So with that said, first off, we need to note that we are told that X is an element of, so that's what that curly looking E shape is. So X is an element of parentheses negative infinity -1 square bracket, and let's take a moment here to pause to note that a square bracket indicates a closed interval, whereas a parenthesis indicates an open interval. And we also need to note that X is an element of square bracket 0.1 in square brackets. Fantastic. So now we need to note once again that X to the power of 5 is less than or equal to F of X, which is less than or equal to x to the power of 3, and we also need to note that X is an element of square bracket 1.0. And that X is an element of square bracket 1.com. Positive infinity, and this is for the condition where X to the power of 3 is less than or equal to F of X and F of X is less than or equal to X to the power of 5. So we need to notice that the bounds are reversed on the different intervals, but in each case, the bounds are X to the power of 3 and X to the power of 5. So, as we should recall and note, the squeeze or sandwich theorem will guarantee that the limit as X approaches C of F of X will equal L. If the limit as X approaches C of the power of 3 is equal to the limit as X approaches C of X 5 equals L, specifically capital L. So, therefore, as a result, we need to find the points C. Where C to the power of 3 is equal to C to the power of 5. So with that said, we need to take C of 5 minus C 3 is equal to 0. And when we do a little bit of maneuvering and shifting things around, we'll find that C to the power of 3, multiplied by C2 minus 1 is equal to 0. So this implies either that C 3 is equal to 0, or C 2 minus 1 is equal to 0. So therefore the solutions will be C 3 equals 0 or C is equal to 0. And this will also mean that C2 minus 1 is equal to 0, and this will mean that C is equal to 1 or C is equal to -1. So now we need to compute the limits at these points. So when C is equal to -1, That will mean that when we plug in a value of -1 in for C, so -1 to the 3 equals -1. And -1 to the power of 5 equals -1. Fantastic. So -1 to the power of 5 is equal to -1. So by squeeze theorem, that will mean that the limit as X approaches -1, F of X will be equal to -1. So now we need to shift gears ever so slightly and focus at C equals 0. So at C equals 0, that will mean that 0 to the power of 3 is equal to 0 and 0 to 5 will be equal to 0. So, therefore, as a result, the limit as X approaches 0 of F of X will be equal to 0, and finally at C equals 1. That will mean that 1 of 3 is equal to 1 and 15 is equal to 1. So that will mean that the limit as X approaches 1 of F of X will be equal to 1. So therefore, without a doubt, the limits are known enforced by the bounds only at the points C equals curly bracket -101. So, going back up to the top quickly, that will mean that our final answers, without a doubt, will have to most definitely be the following. So let's quickly recap our final answers. So C equals curly brackets, -101, curly bracket. And our limits will be the limit as X approaches -1 of F of X is equal to the limit as X approaches -1 X 3 is equal to the limit as X approaches -1 of x 5 is equal to -1. And the limit as X approaches 0 of F of X is equal to the limit as X approaches 0 of X 3 is equal to the limit as X approaches 0 of X of 5, which is all equal to 0. And finally, the limit as X approaches 1 of F of X is equal to the limit as X approaches 1 of X 3 is equal to the limit as X approaches 1 of X to the power of 5 is equal to 1. And that's it. We've officially solved for this problem. Hooray. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Theory and Examples

If x⁴ ≤ f(x) ≤ x² for x in [−1,1] and x² ≤ f(x) ≤ x⁴ for x < - 1 and x > 1, at what points c do you automatically know limx→c f(x)? What can you say about the value of the limit at these points?

Key Concepts

Limit of a Function

Squeeze Theorem

Continuity of Functions

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