# How do you evaluate the limit #lim((x-3)^2-9)/(2x)dx# as #x->0#?

The limit equals

First expand.

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To evaluate the limit lim((x-3)^2-9)/(2x)dx as x approaches 0, we can directly substitute the value of x into the expression. Plugging in x=0, we get (0-3)^2-9)/(2*0), which simplifies to (-3)^2-9)/(0). Simplifying further, we have (9-9)/(0), which equals 0/0. This is an indeterminate form, so we need to apply a different method to evaluate the limit. One approach is to factorize the numerator as (x-3)^2-9 = (x-3+3)(x-3-3) = (x)(x-6). Rewriting the expression, we have lim((x)(x-6))/(2x) as x approaches 0. Canceling out the common factor of x, we get lim(x-6)/2 as x approaches 0. Plugging in x=0, we have (0-6)/2, which simplifies to -6/2 = -3. Therefore, the limit of ((x-3)^2-9)/(2x) as x approaches 0 is -3.

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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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