Trigonometric limits & squeeze theorem
Squeeze theorem example
The graphs of f of x, g of x, and h of x are shown below. Select and drag cards to create a compound inequality that orders the values of f of x, g of x, and h of x for x-values near 2 but not at 2 itself. So for any of the x-values that are depicted right over here, say, x is equal to 3, we see that h of 3 is the largest, f of 3 is the smallest, and g of 3 is in between. And that's true for any of the x-values that they've depicted here. If we look at when x is equal to 1, h of 1 is the largest, f of 1 is the smallest, g of 1 is in between. So for all of the x-values that they're depicting, f of x is less than or equal to g of x, which is less than or equal to h of x. And the only place where they equal based on this graph comes into play, it looks like as we approach x equals 2, it looks like all the functions are approaching 1. So that's where the equal might come into play. But let's keep seeing what they want us to do after that. So then it says it follows that. And instead of writing f of x, g of x, and h of x, they've written the actual definitions of them. So let's just remind ourselves f of x is 2x times the square root of x minus 1 minus 1. That's this blue one right over here. So instead writing f of x, we can write 2 times the square root of x minus 1. Minus 1 is less than or equal to g of x. G of x was this rational expression right over here. So let's go back down here. We get this rational expression. And then that's going to be less than or equal to h of x, which was, I believe, e to the x minus 2. Is that right? Yep, e to the x minus 2. So all we've really done is replaced f, g, and h with their definitions. And then this means that the limit-- so they're looking at the limit as x approaches 2 of these different expressions. So the limit as x approaches 2 of this expression is going to be less than or equal to the limit as x approaches 2 of this expression, which is this right over here, which is going to be less than or equal to the limit as x approaches 2 of this expression, which is that right over there. And then they say finally the value of the limit as x approaches 2 of this thing right over here is-- well, this is where the squeeze theorem comes into play. We just have to remind ourselves-- well, let's just think about it. Can we figure out the limit as x approaches 2 of this right over here? Well, the limit as x approaches 2-- let's see, 2 minus 1. So we're taking the principal root of 2 minus 1, which is the principal root of 1. So you have 2 times 1 minus 1. So this is 1. This right over here is e to the 2 minus 2. That's either the 0, or that's 1 as well. So the limit of all of this is going to be greater than or equal to 1, and it's going to be less than or equal to 1. Or it's right in between 1 and 1. And the only way that it's going to be between 1 and 1 is if it is equal to 1. This is the squeeze theorem at play right over here. g of x, over the domain that we've been looking at, or over the x-values that we care about-- g of x was less than or equal to h of x, which was-- or f of x was less than or equal to g of x, which was less than or equal to h of x. And then we took the limit for all of them as x approached 2. For the lower function, for f of x, it approached 1. And we see it in the graph right over here. In the lower function, f of x approaches 1, h of x approaches 1, and therefore, g of x must also approach 1. And we actually see that in this graph right over here. But anyway, we could check our answer just to feel good about ourselves. We got it right.