How do you evaluate the limit #(sqrt(x+2)-sqrt(2-x))/x# as x approaches #0#?
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To evaluate the limit (sqrt(x+2)-sqrt(2-x))/x as x approaches 0, we can use algebraic manipulation and the limit properties. First, we rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator, which is (sqrt(x+2)+sqrt(2-x)). Simplifying this expression gives us (x+2) - (2-x) in the numerator. Expanding and canceling terms, we get 4x in the numerator. The denominator remains as x. Now, we can cancel out the common factor of x. Taking the limit as x approaches 0, we are left with 4/1, which simplifies to 4. Therefore, the limit of (sqrt(x+2)-sqrt(2-x))/x as x approaches 0 is equal to 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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