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Use Definition 9 to prove that $ \displaystyle \lim_{x \to \infty} e^x = \infty $.

Given $M>0,$ we need $N>0$ such that $x>N \Rightarrow e^{x}>M .$ Now $e^{x}>M \Leftrightarrow x>\ln M,$ so take

$N=\max (1, \ln M) . \text { (This ensures that } N>0 .)$ Then $x>N=\max (1, \ln M) \Rightarrow e^{x}>\max (e, M) \geq M$

so $\lim _{x \rightarrow \infty} e^{x}=\infty$

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This is problem number seventy nine of the Stuart Calculus eighth edition, Section two point six. Use definition nine. To prove that limit is expertise infinity of either the X is equal to infinity. And our definition nine states that for every m, we can find an end touch that if an X is greater than I am, the function will definitely be greater than the value em. So again, we want to confirm that for every m there exists and then such that it makes this truth em and and being restricted to positive values. From the second part here, our function is e to the X greater than I am and we're looking forward. And that corresponds to this. So take the natural log of both sides. I rely on you to the ex greater than the natural log of them here, the national either the excess just x creator than Ellen of em. What we have here now we can compare with this conditional here X is greater than end. And our choice for end just needs to be at least Ellen of M. So this one example that definitely works if we choose and to be element. For example, if we choose acts to be greater than end where we choose end to be Ellen of Emma, we can go ahead and do what we just did before. Eyes take the exponential to both sides are the opposite of what we did. Exponential of Ellen. Mm. The right side reduces to just em. Andi, this is exactly the condition and we need here. Ethics is equal to X, and it's greater than him. So a choice like and is equal to Alan M. Is appropriate to prove our statement. We can choose any value of and given a nem value, and this is exactly what the dimension states. So by definition, and we have proven that limit is expressions infinity of GTX. It's definitely eat. It's definitely infinity.