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How do you find the limit of #(sqrt(x+4) -2) / x# as x approaches 0?

Answer 1

Elijah Abram

#1/4#

We have limit of indeterminate form, ie #0/0# so can use L'Hopital's rule:

#lim_(xrarr0) (sqrt(x+4) - 2)/x = lim_(xrarr0)(d/(dx)(sqrt(x+4)-2))/(d/(dx)(x))#

#=lim_(xrarr0)(1/(2sqrt(x+4)))/1 = 1/(2sqrt(0+4)) = 1/4#

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Answer 2

Emilia Aguilar

To find the limit of (sqrt(x+4) -2) / x as x approaches 0, we can use algebraic manipulation and the concept of limits.

First, let's simplify the expression by multiplying both the numerator and denominator by the conjugate of the numerator, which is (sqrt(x+4) + 2). This will help us eliminate the square root in the numerator.

[(sqrt(x+4) - 2) / x] * [(sqrt(x+4) + 2) / (sqrt(x+4) + 2)]

Simplifying this expression gives us:

[(x+4) - 4] / (x * (sqrt(x+4) + 2))

Further simplifying, we get:

(x) / (x * (sqrt(x+4) + 2))

Now, we can cancel out the x terms:

1 / (sqrt(x+4) + 2)

As x approaches 0, the expression becomes:

1 / (sqrt(0+4) + 2) = 1 / (2 + 2) = 1 / 4

Therefore, the limit of (sqrt(x+4) -2) / x as x approaches 0 is 1/4.

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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.