How do you find the Limit of #sqrt(x^2 - 9) / (x - 3) # as x approaches 3?

Answer 1

Always try substitution first. (it won't work for this one.) When finding a limit of a fraction and in doubt, rationalize either the numerator or denominator.

#sqrt(x^2-9)/(x-3)#

If we rationalize the numerator, we'll be able to factor and reduce, so that looks reasonable.

#sqrt(x^2-9)/(x-3) * sqrt(x^2-9)/(sqrt(x^2-9)) = (x^2-9)/((x-3)sqrt(x^2-9))#
# = ((x-3)(x+3))/((x-3)sqrt(x^2-9))#
# = (x+3)/sqrt(x^2-9)#
Now, as #x# approaches #3#, it must be on the left. (I am assuming that we want to stay in the real numbers.)
As #xrarr3^-#, the numerator is approaching #6# and the denominator is a positive number approaching #0#.
As #xrarr3^-#, the ratio is increasing without bound.
#lim_(xrarr3^-) sqrt(x^2-9)/(x-3) = lim_(xrarr3^-) (x+3)/sqrt(x^2-9) =oo #
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Answer 2

To find the limit of sqrt(x^2 - 9) / (x - 3) as x approaches 3, we can simplify the expression by factoring the numerator. The expression can be rewritten as (x + 3) / (x - 3).

Now, we can substitute x = 3 into the simplified expression to find the limit. However, this would result in division by zero, which is undefined.

To overcome this issue, we can use algebraic manipulation. By factoring the numerator as (x + 3), we can cancel out the common factor of (x - 3) in the denominator.

After canceling out the common factor, we are left with the expression 1 / 1, which equals 1.

Therefore, the limit of sqrt(x^2 - 9) / (x - 3) as x approaches 3 is 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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