How do you find the limit #(2-sqrt(x+2))/(4-x^2)# as #x->2#?

Answer 1

I got: #1/16#

If you try to use #2# directly you get the indeterminate form #0/0#; we can try using De L'Hospital Rule deriving top and bottom and then apply the limit: #lim_(x->2)((2-sqrt(x+2))/(4-x^2))=lim_(x->2)((-1/(2sqrt(x+2)))/(-2x))=# as #x->2# we get: #lim_(x->2)((-1/(2sqrt(x+2)))/(-2x))=1/16#
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Answer 2

To find the limit of (2-sqrt(x+2))/(4-x^2) as x approaches 2, we can substitute the value of 2 into the expression. However, this would result in division by zero, which is undefined. Therefore, we need to simplify the expression before substituting the value of 2.

First, we can simplify the numerator by multiplying both the numerator and denominator by the conjugate of the numerator, which is (2+sqrt(x+2)). This will help eliminate the square root.

After simplifying, we get (2-sqrt(x+2))/(4-x^2) = [(2-sqrt(x+2))(2+sqrt(x+2))]/[(4-x^2)(2+sqrt(x+2))].

Next, we can simplify the denominator by factoring it as a difference of squares: (4-x^2) = (2-x)(2+x).

Now, we can cancel out the common factors in the numerator and denominator, which is (2-x).

After canceling, we are left with (2+sqrt(x+2))/(2+x).

Finally, we can substitute the value of x=2 into the simplified expression: (2+sqrt(2+2))/(2+2) = (2+sqrt(4))/4 = (2+2)/4 = 4/4 = 1.

Therefore, the limit of (2-sqrt(x+2))/(4-x^2) as x approaches 2 is equal to 1.

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Answer from HIX Tutor

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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