# Lim x -> 0 of (1/(2+x) - 1/2)/x

• IntegrateMe
This gives us a = -1/4 and b = 1/2.In summary, you can solve the limit of [1/(2+x) - 1/2] / x without using L'Hopital's rule by using partial fractions and finding the solution of -1/4 when x goes to 0.
IntegrateMe
lim x tends to 0 of:

[1/(2+x) - 1/2] / x

I simplified the fraction and got:

[x / 2(2+x)] / x

and with further simplification:

x / x(4+2x)

How do I solve this without using L'Hopital's rule?

The answer is 1, by the way.

You can use "partial fractions" to simplify.
[1/(2+x) - 1/2] / x <=> 1/x(2+x) - 1/2x <=> 1/2x-1/(2x+4)-1/2x <=> -1/(2x+4)

When x goes to 0, the solution is goes towards -1/4 = -1/(2*0+4)

The partial fraction of found by solving
1/x(2+x) = a/x + b/(x+2)

You can use "partial fractions" to simplify.
[1/(2+x) - 1/2] / x <=> 1/x(2+x) - 1/2x <=> 1/2x-1/(2x+4)-1/2x <=> -1/(2x+4)

When x goes to 0, the solution is goes towards -1/4 = -1/(2*0+4)

The partial fraction of found by solving
1/x(2+x) = a/x + b/(x+2)

## 1. What is the limit of the expression (1/(2+x) - 1/2)/x as x approaches 0?

The limit of the expression is equal to 1/4.

## 2. How do you solve for the limit of (1/(2+x) - 1/2)/x as x approaches 0?

To solve for the limit, we can simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator, which is (2+x). This will eliminate the denominator's x term and allow us to evaluate the limit easily.

## 3. Can the limit of (1/(2+x) - 1/2)/x be evaluated by direct substitution?

No, direct substitution cannot be used to evaluate the limit as it would result in an indeterminate form (0/0).

## 4. What is the significance of the limit of (1/(2+x) - 1/2)/x as x approaches 0 in calculus?

This limit is significant in calculus as it represents the derivative of the function 1/(2+x), which can be written as f'(x).

## 5. How can the limit of (1/(2+x) - 1/2)/x be used in real-world applications?

The limit can be used to determine the instantaneous rate of change of a function, which is useful in fields such as physics, engineering, and economics. It can also be used to calculate the slope of a tangent line to a curve at a specific point.

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