For $\lim _ { x \rightarrow 2 } x ^ { 3 } = 8$ and $\varepsilon = 0.2$ , approximate the largest value of $\delta$ that satisfies the definition of the limit.
A) 0.0011
B) 0.0012
C) 0.01
D) 0.001

Question

Quizplus · Accepted Answer

To satisfy the limit definition, we need $|x^3 - 8| < 0.2$. Approximating around $x = 2$, we use the derivative $3x^2$ to estimate $\delta$. At $x = 2$, $3x^2 = 12$, so $\delta \approx \frac{0.2}{12} \approx 0.0167$. The closest option is 0.01.

For lim⁡x→2x3=8\lim _ { x \rightarrow 2 } x ^ { 3 } = 8limx→2​x3=8

For $\lim _ { x \rightarrow 2 } x ^ { 3 } = 8$