Answer:
[tex]\lim_{x \to 0} \frac{\frac{1}{6+x}-\frac{1}{6}}{x}=-1/36[/tex]
Step-by-step explanation:
So we have the limit:
[tex]\lim_{x \to 0} \frac{\frac{1}{6+x}-\frac{1}{6}}{x}[/tex]
Let's remove the fractions in the denominator by multiplying both layers by (6+x)(6). So:
[tex]\lim_{x \to 0} \frac{\frac{1}{6+x}-\frac{1}{6}}{x}\cdot (\frac{(6+x)(6)}{(6+x)(6)})[/tex]
Distribute:
[tex]=\lim_{x \to 0} \frac{(6)-(6+x)}{x(6+x)(6)}[/tex]
Simplify the numerator:
[tex]=\lim_{x \to 0} \frac{6-6-x}{x(6+x)(6)}\\=\lim_{x \to 0} \frac{-x}{x(6+x)(6)}[/tex]
Both the numerator and the denominator have an x. Cancel:
[tex]=\lim_{x \to 0} \frac{-1}{(6+x)(6)}[/tex]
Direct substitution:
[tex]= \frac{-1}{(6+0)(6)}[/tex]
Simplify:
[tex]=-1/36[/tex]
And that's our answer.
And we're done!
