Take it to the limit

Calculus Level 3

Given $f\left( x \right) = \frac{x + 5}{2}$ , $g\left( x \right) = x + 7$ , $\lim_{x \rightarrow 6}{f\left( g\left( x \right)\right)} = L$ and that $\lvert f\left( g\left( x \right)\right) - L \rvert < 0.005$ , what is the value of $\delta$ such that $\lvert x - 6 \rvert < \delta$ ?

Detail : all the functions' domains and codomains, in this problem, are the set of the Real Numbers.

The answer is 0.01.

1 solution

Bruno Oggioni Moura
Jan 16, 2016

$f\left( g\left( x \right)\right) = \frac{(x + 7) + 5}{2} = \frac{x}{2} + 6$ . Noticing that $\lim_{x \rightarrow 6}{f\left( g\left( x \right)\right)} = 9$ , we change the values of the given inequality:

$\lvert f\left( g\left( x \right)\right) - L \rvert < 0.005 \Rightarrow \lvert (\frac{x}{2} + 6) - 9 \rvert < 0.005 \Rightarrow \lvert \frac{x - 12 - 18}{2} \rvert < \frac{0.01}{2} \Rightarrow \lvert x - 6 \rvert < 0.01 \Rightarrow \delta = 0.01$

Take it to the limit

The answer is 0.01.

1 solution

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