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Lim x+1/( X->1 sqrt(x^3+8)-3)

Mathematics
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I’m not sure whether you mean x + [1/(sqrt(x^3 + 8) – 3)] or if you mean (x + 1)/(sqrt(x^3 + 8) – 3). Neither of them has a limit. This is because the left and right hand limits aren't the same. I’ll show that (x + 1)/(sqrt(x^3 + 8) – 3) has no limit as x goes to 1. If you meant the first expression, the reasoning is the same. Imagine approaching 1 from the left hand side. Then sqrt(x^3 +8) is always a little less than 3. So sqrt(x^3 +8) – 3 is always negative as we let x approach 1 from the left side. But notice that sqrt(x^3 +8) – 3 gets really small as x goes to 1. So that means 1/[sqrt(x^3 +8) – 3] goes to negative infinity as x goes to 1 from the left side. Now imagine approaching 1 from the right hand side. Then sqrt(x^3 +8) is always a little bigger than 3. So sqrt(x^3 +8) – 3 is always positive as we let x approach 1 from the right side. But notice that sqrt(x^3 +8) – 3 gets really small as x goes to 1. So that means 1/[sqrt(x^3 +8) – 3] goes to positive infinity as x goes to 1 from the left side. We also know that lim (x + 1) as x goes to 1 is just 2. So our expression, numerator is bounded while the denominator gets infinitely small. So lim (x + 1)/(sqrt(x^3 + 8) – 3) equals - infinity as x approaches from the left. And lim (x + 1)/(sqrt(x^3 + 8) – 3) equals + infinity as x approaches from the right. So there is no limit since the left and right limits aren’t equal.

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