For $\lim _ { x \rightarrow 3 } x ^ { 2 } = 9$ and $\varepsilon = 0.35$ , approximate the largest value of $\delta$ that satisfies the definition of the limit.
A) 0.04
B) 0.05
C) 0.06
D) 0.03

Question

Quizplus · Accepted Answer

Given $\lim _ { x \rightarrow 3 } x ^ { 2 } = 9$ and $\varepsilon = 0.35$, we want to find $\delta$ such that if $0 < |x - 3| < \delta$, then $|x^2 - 9| < 0.35$. Simplifying $|x^2 - 9| < 0.35$ gives $8.65 < x^2 < 9.35$. Taking square roots, we find that approximately $2.94 < x < 3.06$, so $|x - 3| < 0.06$. However, to ensure $|x^2 - 9| < 0.35$, we choose the largest $\delta$ less than 0.06, which is 0.05.

For lim⁡x→3x2=9\lim _ { x \rightarrow 3 } x ^ { 2 } = 9limx→3​x2=9

For $\lim _ { x \rightarrow 3 } x ^ { 2 } = 9$