# How do you find the limit of #(sqrt(1+2x))-(sqrt(1-4x)] / x# as x approaches 0?

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To find the limit of the given expression as x approaches 0, we can use algebraic manipulation and the limit properties.

First, let's simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator, which is (sqrt(1+2x))+(sqrt(1-4x)). This will help us eliminate the square roots in the numerator.

[(sqrt(1+2x))-(sqrt(1-4x))] * [(sqrt(1+2x))+(sqrt(1-4x))] / [x * (sqrt(1+2x))+(sqrt(1-4x))]

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

[(1+2x) - (1-4x)] / [x * (sqrt(1+2x))+(sqrt(1-4x))]

Simplifying further, we have:

[3x] / [x * (sqrt(1+2x))+(sqrt(1-4x))]

Now, we can cancel out the common factor of x:

[3] / [(sqrt(1+2x))+(sqrt(1-4x))]

As x approaches 0, the expression inside the square roots becomes 1, and the limit becomes:

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

Simplifying further, we have:

[3] / [2]

Therefore, the limit of the given expression as x approaches 0 is 3/2.

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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.

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