Question

Given a function f, defined on R by
f(x) = x^2/(x^2 + 4) ,
l= 1 and €= 0.1, find k > 0
such that x > k => |f(x) - 1| < €.

Answer 1

Answer on Question #83137 – Math – Calculus

Question

Given a function $f$, defined on $R$ by $f(x) = x^{2} / (x^{2} + 4)$, $l = 1$ and $\varepsilon = 0.1$, find $k > 0$ such that $x > k \Rightarrow |f(x) - 1| < \varepsilon$.

Solution

We have $f(x) - 1 = (x^{2} - (x^{2} + 4)) / (x^{2} + 4) = -4 / (x^{2} + 4)$, so that $|f(x) - 1| = 4 / (x^{2} + 4)$. The inequality $|f(x) - 1| < \varepsilon$ then reads $4 / (x^{2} + 4) < \varepsilon$ and has solution $x^{2} > 4 / \varepsilon - 4$. Substituting $\varepsilon = 0.1$, we get $x^{2} > 36$, hence $|x| > 6$.

Answer: $k = 6$.

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