5

8 Determine the following limits. If the limits doesn 't exist, determine wther it is 0, ~0 or neither. 2x + 5 lim 17-2 x + 2 (z +3)32 + 1) (b) lim +0O (x - 1)...

Question

8 Determine the following limits. If the limits doesn 't exist, determine wther it is 0, ~0 or neither. 2x + 5 lim 17-2 x + 2 (z +3)32 + 1) (b) lim +0O (x - 1)2 Vx4 2x2 +8 lim 1--2 2x 2x - 1 (d) lim I +0.5 12x3 x2lim 1-0c2 + x cos 5x _ 1 lim 1-0 3x 15 sin(€ + 2) lim 1--2 c2 _ I _ 6

8 Determine the following limits. If the limits doesn 't exist, determine wther it is 0, ~0 or neither. 2x + 5 lim 17-2 x + 2 (z +3)32 + 1) (b) lim +0O (x - 1)2 Vx4 2x2 +8 lim 1--2 2x 2x - 1 (d) lim I +0.5 12x3 x2 lim 1-0 c2 + x cos 5x _ 1 lim 1-0 3x 15 sin(€ + 2) lim 1--2 c2 _ I _ 6



Answers

In each part, find the limit by inspection.
(a) $\lim _{x \rightarrow 8} 7$ (b) $\lim _{x \rightarrow-\infty}(-3)$ (c) $\lim _{x \rightarrow 0^{+}} \pi$ (d) $\lim _{x \rightarrow-2} 3 x$ (e) $\lim _{n \rightarrow \infty} 12 y$ (f) $\lim _{h \rightarrow+\infty}(-2 h)$

For the following problem, we want to um determine the limits analytically. So we see that the limit as we approach two From the right of one over X -2. So doing this analytically and geographically we're approaching two from the right, meaning that this is going to be a number larger than two, such as um three 2.1, But regardless this is going to be a positive number, but just a very, very small positive number. So we have one over a very, very small positive number. That means that we're going to get a large number, a large positive number. So this is going to be infinity. Okay, Then, using this logic with the other one as we approach from the left, we're going to get a very, very small negative number, meaning this is going to become negative infinity. So then the overall limit l we see will not exist because um we get to different limits from the two different sites.

For the given problem, we're going to consider the limit as X approaches -2 from the left and right of X -4 over X times X plus two. And we're going to be approaching a negative two from the right. 1st Approaches -2 from the right means that we have um smaller a smaller negative value. So it's going to mean that this value is positive but this value right here is automatically negative. So this whole value here is negative and then this value here is negative. So we get a negative over negative that's going to be um positive. So we get infinity as we approach from the left then as we approach from the right or sorry as we approach them right as we approach from the left this value is going to be negative, this value is going to be negative. That makes this value positive and then this here is going to be um negative. So we get a negative over positive meaning it's negative but this value is gonna be extremely close to zero, meaning this is going to be a negative infinity. That's going to be our final answer. So the limit does not exist

As next approaches to from the positive direction it's considered a fraction one over X minus two one, of course, approaches one. It's always one and X minus. Two approaches zero From the positive side, that means that the whole thing the entire fraction approaches positive infinity as X approaches two from the negative side, our Fraction one over X minus two approaches one over zero from the negative side in this form really means negative infinity, the 21 handed limits being different. We have no limit as X approaches to before one over X minus two. So the answer to a it's positive infinity. The answer to Part B is negative Infinity and the answer to part seeing is no limit exists.

Let's start with a you notice here that we have a condom in front sort of limit as experts to from a rights that's this equal to one and then on our denominator if we plug in, executed to get so so we know that this approach is they're all. And since we have two from our right hand side, so it's like 2.1 and when it's to get a positive value so we have positive and it approaches zero. So we know that, um, we have one over a positive approaching zero. So this is equal to, um, positive infinity. I notice for part B. Again, we have one. We have our function X minus two. It approaches girl. And since we have to from the left outside, we have a value such as 1.91 point nine minutes to that's equal to a negative value. So that's negative and are approaching Darryl in this case, we're approaching a negative. Infinity, I notice that's are left handed on the right hand of the myth are not the same values. Voter limits as expected to does not exist


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