Question

I'm stuck on this question, any ideas? thanks

Answer 1

-1/36

Explanation:

lim(x→0) [1/(6 + x) − 1/6] / x

The common denominator of the two fraction in the numerator is 6(6 + x) = 36 + 6x.

lim(x→0) [6/(36 + 6x) − (6 + x)/(36 + 6x)] / x

Combine the fractions.

lim(x→0) [(6 − 6 − x) / (36 + 6x)] / x

Simplify.

lim(x→0) [-x / (36 + 6x)] / x

Divide.

lim(x→0) -1 / (36 + 6x)

Evaluate.

-1/36

Answer 2

`\lim_(x \to 0) ((1)/(6+x)-(1)/(6))/(x)=-1/36`

Explanation:

So we have the limit:

`\lim_(x \to 0) ((1)/(6+x)-(1)/(6))/(x)`

Let's remove the fractions in the denominator by multiplying both layers by (6+x)(6). So:

`\lim_(x \to 0) ((1)/(6+x)-(1)/(6))/(x)\cdot (((6+x)(6))/((6+x)(6)))`

Distribute:

`=\lim_(x \to 0) ((6)-(6+x))/(x(6+x)(6))`

Simplify the numerator:

`=\lim_(x \to 0) (6-6-x)/(x(6+x)(6))\\=\lim_(x \to 0) (-x)/(x(6+x)(6))`

Both the numerator and the denominator have an x. Cancel:

`=\lim_(x \to 0) (-1)/((6+x)(6))`

Direct substitution:

`= (-1)/((6+0)(6))`

Simplify:

`=-1/36`

And that's our answer.

And we're done!