I need help with number 7 !! Sorry for bad picture quality 5516 00 PM iPad x htt

Question

I need help with number 7 !! Sorry for bad picture quality 5516 00 PM iPad x https://www.web xta https://www.che ×'D Larson, Calculus·D Larson Calculus ., d c https://www webassign net/web/StudentAssignment i 0 The limit exists at points on the graph except where c-4. OThe Smit exists at all points on the graph except where e 2 and e-4. O The limit exists at all points on the graph O The lime exists at a. ports on the graph except where c-2. Consider th, following limit tim (+ 4) Find the limit L Find E>such tet Inn-u-001 whenever

Explanation / Answer

|X - 4| < delta

-delta < x - 4 < delta

4 - delta < x < 4 + delta

We want to prove that |x^2 - 16| < epsilon

The above means the following :

-E < x^2 - 16 < E

16 - E < x^2 < 16 + E

0 <= x^2 < 16 + E

x^2 - (16 + E) < 0

By using quadratics, the above inequality is true for :

-sqrt(16 + E) < x < sqrt(16 + E)

Comparing with the "delta" inequalities above, let:

4 + D < sqrt(16 + E)

D < sqrt(16 + E) - 4

But here epsilon is 0.01

So, D < sqrt(16 + 0.01) - 4

D < 0.00125

So, delta = 0.00125 ----> ANSWER

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