# How do you find #lim sqrt(u^2-3u+2)-sqrt(u^2+1)# as #u->oo#?

Note that:

Factoring out the terms with the largest degree:

Also note that:

So then:

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To find the limit of the expression lim sqrt(u^2-3u+2)-sqrt(u^2+1) as u approaches infinity, we can simplify the expression by rationalizing the denominator. By multiplying the expression by the conjugate of the denominator, we get:

lim sqrt(u^2-3u+2)-sqrt(u^2+1) * (sqrt(u^2-3u+2)+sqrt(u^2+1)) / (sqrt(u^2-3u+2)+sqrt(u^2+1))

Simplifying further, we have:

lim (u^2-3u+2) - (u^2+1) / (sqrt(u^2-3u+2)+sqrt(u^2+1))

This simplifies to:

lim -3u+1 / (sqrt(u^2-3u+2)+sqrt(u^2+1))

As u approaches infinity, the terms involving u^2 become negligible compared to the -3u term. Therefore, the limit becomes:

lim -3u / (sqrt(u^2)+sqrt(u^2))

Simplifying further, we have:

lim -3u / (u+u)

This simplifies to:

lim -3/2

Therefore, the limit of sqrt(u^2-3u+2)-sqrt(u^2+1) as u approaches infinity is -3/2.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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