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How do you find the limit of #[sqrt(x+1) - 1]/[x]# as x approaches 0?

Answer 1

Aurora Alvarado

#1/2#

#[sqrt(x+1) - 1]/[x] = ((sqrt(x+1)-1)(sqrt(x+1)+1))/(x(sqrt(x+1)+1)=# #(x+1-1)/(x(sqrt(x+1)+1)) = 1/(sqrt(x+1)+1)# so

#lim_(x->0)[sqrt(x+1) - 1]/[x] equiv lim_(x->0) 1/(sqrt(x+1)+1) = 1/2#

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Answer 2

Paisley Johnson

To find the limit of [sqrt(x+1) - 1]/[x] as x approaches 0, we can use algebraic manipulation and the limit properties.

First, let's simplify the expression by multiplying both the numerator and denominator by the conjugate of the numerator, which is sqrt(x+1) + 1. This will help us eliminate the square root in the numerator.

[sqrt(x+1) - 1]/[x] * [sqrt(x+1) + 1]/[sqrt(x+1) + 1]

Expanding the numerator using the difference of squares, we get:

[(x+1) - 1]/[x * (sqrt(x+1) + 1)]

Simplifying further, we have:

[x]/[x * (sqrt(x+1) + 1)]

Now, we can cancel out the x terms:

1/[sqrt(x+1) + 1]

As x approaches 0, the expression becomes:

1/[sqrt(0+1) + 1] = 1/[sqrt(1) + 1] = 1/[1 + 1] = 1/2

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

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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.