(a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state the value of and δ(orMorNandδ) in each case. (i) How close to 4 do we need to take x so that (x2−2)&lt;0.001 ? (ii) How close to 4 do we need to take x so that (x2−2)&gt;0.0001 ? (iii) How close to 0 do we need to take x so that (2x+9)&lt;9.0001 ? (iv) How close to 0 do we need to take x so that (2x+9)&gt;8.999? (v) How close to 0 do we need to take x so that (x2+6x+9)&lt;9.001 ? (vi) How close to 0 do we need to take x so that (x2+6x+9)&lt;9.0001 ? (vii) How close to 0 do we need to take x so that (x2+6x+9)&gt;8.9999 ? (viii) How close to -7 do we need to take x so that 1(x+7)4&gt;10000 ? (ix) How close to −7 do we need to take x so that 1(x+7)4&gt;100000 ? (x) How close to 0 do we need to take x so that ln⁡x&lt;−10000? (b) Use the definition of a limit to show that (i) limx→4 (x2−2)=0 (ii) limx→0 (2x+9)=9 (iii) limx→0 (x2+6x+9)=9 (iv) limx→−7 1(x+7)4=∞ (v) limx→0+ ln⁡x=−∞

Question

(a) Write each answer as an equality, and any decimals up to 7 decimal places. Clearly state the value of and δ(orMorNandδ) in each case. (i) How close to 4 do we need to take x so that (x2−2)&amp;lt;0.001 ? (ii) How close to 4 do we need to take x so that (x2−2)&amp;gt;0.0001 ? (iii) How close to 0 do we need to take x so that (2x+9)&amp;lt;9.0001 ? (iv) How close to 0 do we need to take x so that (2x+9)&amp;gt;8.999? (v) How close to 0 do we need to take x so that (x2+6x+9)&amp;lt;9.001 ? (vi) How close to 0 do we need to take x so that (x2+6x+9)&amp;lt;9.0001 ? (vii) How close to 0 do we need to take x so that (x2+6x+9)&amp;gt;8.9999 ? (viii) How close to -7 do we need to take x so that 1(x+7)4&amp;gt;10000 ? (ix) How close to −7 do we need to take x so that 1(x+7)4&amp;gt;100000 ? (x) How close to 0 do we need to take x so that ln⁡x&amp;lt;−10000? (b) Use the definition of a limit to show that (i) limx→4 (x2−2)=0 (ii) limx→0 (2x+9)=9 (iii) limx→0 (x2+6x+9)=9 (iv) limx→−7 1(x+7)4=∞ (v) limx→0+ ln⁡x=−∞

Raheem Donnelly · Accepted Answer

Step 1 &quot;Since you have posted a question with multiple sub-parts, we will solve first four sub-parts for you. To get remaining sub-part to be solved please re-post the complete question and mention the sub-parts to be solved&quot;. a. (i) How close to 4 do we need to take x so that (x2−2)&amp;lt;0.001 ? Let the inequality be x2−2&amp;lt;0.001 x2−2&amp;lt;0.001 ⇒x2&amp;lt;0.001+2 ⇒x&amp;lt;2.001(2) ⇒x&amp;lt;4.002 x should be close to 4 by 0.002 so that x2−2&amp;lt;0.001 Step 2 (ii) How close to 4 do we need to take x so that (x2−2)&amp;gt;0.0001 ? Let the inequality be x2−2&amp;gt;0.0001 x2−2&amp;gt;0.0001 ⇒x2&amp;gt;0.0001+2 ⇒x&amp;gt;2.0001(2) ⇒x&amp;gt;4.0002 x should be close to 4 by 0.0002 so that x2−2&amp;gt;0.0001 Step 3 (iii) How close to 0 do we need to take x so that (2x+9)&amp;lt;9.0001 ? Let the inequality be (2x+9)&amp;lt;9.0001 (2x+9)&amp;lt;9.0001 ⇒2x&amp;lt;9.0001−9 ⇒2x&amp;lt;0.0001 ⇒x&amp;lt;0.00005 x should be close to 0 by 0.00005 so that (2x+9)&amp;lt;9.0001 Step 4 (iv) How close to 0 do we need to take x so that (2x+9)&amp;gt;8.999? Let the inequality be (2x+9)&amp;gt;8.999 (2x+9)&amp;gt;8.999 ⇒2x&amp;gt;8.999−9 ⇒2x≻0.001 ⇒x≻0.0005 x should be close to 0 by - 0.0005 so that (2x+9)&amp;gt;8.999

(a) Write each answer as an equality, and any decimals up to 7 decimal places. C

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