Question

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

Answer 1

Answer:

$\lim_{x \to 0} \frac{\frac{1}{6+x}-\frac{1}{6}}{x}=-1/36$

Step-by-step explanation:

So we have the limit:

$\lim_{x \to 0} \frac{\frac{1}{6+x}-\frac{1}{6}}{x}$

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

$\lim_{x \to 0} \frac{\frac{1}{6+x}-\frac{1}{6}}{x}\cdot (\frac{(6+x)(6)}{(6+x)(6)})$

Distribute:

$=\lim_{x \to 0} \frac{(6)-(6+x)}{x(6+x)(6)}$

Simplify the numerator:

$=\lim_{x \to 0} \frac{6-6-x}{x(6+x)(6)}\\=\lim_{x \to 0} \frac{-x}{x(6+x)(6)}$

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

$=\lim_{x \to 0} \frac{-1}{(6+x)(6)}$

Direct substitution:

$= \frac{-1}{(6+0)(6)}$

Simplify:

$=-1/36$

And that's our answer.

And we're done!

Answer 2

Answer:

-1/36

Step-by-step 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