How do you find the limit #lim_(x->3^+)|3-x|/(x^2-2x-3)# ?
By factoring out the denominator and eliminating the absolute value sign,
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To find the limit of lim_(x->3^+)|3-x|/(x^2-2x-3), we can evaluate the limit from the right side of x=3.
First, let's simplify the expression. The absolute value of (3-x) is equal to (x-3) when x>3. So, we can rewrite the expression as (x-3)/(x^2-2x-3).
Next, we factor the denominator. The expression becomes (x-3)/[(x-3)(x+1)].
Now, we can cancel out the common factor of (x-3) in the numerator and denominator. This leaves us with 1/(x+1).
Finally, we can take the limit as x approaches 3 from the right side. Plugging in x=3 into the simplified expression, we get 1/(3+1) = 1/4.
Therefore, the limit of lim_(x->3^+)|3-x|/(x^2-2x-3) is 1/4.
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To find the limit (\lim_{x \to 3^+} \frac{|3 - x|}{x^2 - 2x - 3}), first consider values of (x) approaching 3 from the right side ((x \to 3^+)).
As (x) approaches 3 from the right side, the expression (|3 - x|) becomes (|3 - 3|) which equals 0.
Also, as (x) approaches 3 from the right side, the expression (x^2 - 2x - 3) approaches (3^2 - 2(3) - 3 = 0).
Therefore, when evaluating the limit, you have (0/0), which is an indeterminate form.
To evaluate further, factorize the denominator (x^2 - 2x - 3 = (x - 3)(x + 1)).
Now, you can simplify the expression:
[ \lim_{x \to 3^+} \frac{|3 - x|}{x^2 - 2x - 3} = \lim_{x \to 3^+} \frac{0}{(x - 3)(x + 1)} = \lim_{x \to 3^+} \frac{0}{0} ]
Since you still have an indeterminate form, you can apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to (x).
After applying L'Hôpital's Rule, you get:
[ \lim_{x \to 3^+} \frac{|3 - x|}{x^2 - 2x - 3} = \lim_{x \to 3^+} \frac{-1}{2x - 2} ]
Now, plug in (x = 3) into the expression to find the limit:
[ \lim_{x \to 3^+} \frac{-1}{2x - 2} = \frac{-1}{2(3) - 2} = \frac{-1}{4} ]
Therefore, the limit (\lim_{x \to 3^+} \frac{|3 - x|}{x^2 - 2x - 3}) is (-\frac{1}{4}).
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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.
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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