Problem 42 Medium Difficulty

Prove, using Definition 6, that $ \displaystyle \lim_{x \to -3} \frac{1}{(x + 3)^4} = \infty $.

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this's problem or forty two of the Stuart Calculus, either edition section two point four prove using definition six. That the limit is experts is negative. Three of the function one over the quantity X plus three to the fourth power. It's called an identity. So definition six states that for any m greater than zero there exists a delta grid. Isn't zero such that if the absolute value of the difference between X and me is Liston daughter, then the function is guaranteed to be great with them. So we just being with second inequality and see what relationship we confined between them and Delta. So our function this one over the quantity X plus three to the fourth and we want to guarantee that this greater than any value we choose for them. We re arrange this one over him created than the quantity express T to the fourth. And then we take ah, the forth route to put science And here the forth route cancel. And what we're left with is that the quantity extra story must be less Stan the must be listen one over the forth route of him. If we recall the conditional for Delta Is that expectancy or in this case, Xmas Negative three. Must be. This difference was fearless and Delta and And we re write it this way. You can see that Delta and this term here. Ah, having value equal to that is an appropriate choice that guarantees bad for any delta. Are you can find any delta for any given value? Mm. For example. Mmm is ten thousand. The four three ten thousand is ten. So choosing the delta equal to one over ten. Ah provide prevents the conditions needed to prove this limit. So for any value, them there exists a daughter, and therefore the positive values and the both correspond to these inequalities. And everything is consistent there forthis summit as expressing it. Three of dysfunction, unequal to infinity.