Determine the following limits.

c. lim x→−2 (...

Question

Determine the following limits.

c. lim x→−2 (x − 4) / x(x + 2)

Accepted Answer

Welcome back, everyone. Evaluate the limit. Limit as x approaches -1 of x + 5 divided by X2 + X. We're given 4 answer choices A says 0, B negative infinity, C infinity, and D does not exist. So let's understand this limit. First of all, let's attempt direct substitution. We get limit as x approaches -1 of x + 5 divided by x2 + x. If we apply direct substitution, we're going to get -1 + 5 divided by -12 + -1. And this essentially gives us a number divided by 0. We can see that in the numerator we get or in the denominator we get 0. This is a form of infinity. However, to clearly understand what type of infinity we get, we have to violate two limits one from the left and one from the right. If they are equal to each other and they give us the same infinity sign specifically right, then we can conclude that the limit is. Infinity with those signs. But if we get different infinities, for example, if we get negative infinity for one and positive infinity for the other, then the limit does not exist. But it already shows that there is some sort of infinity involved. So what we're going to do first of all is evaluate the limit as X approaches -1 from the left. According to the limit existence COM, we want to check both sides. So the first one is X approaches -1 from the left. And now what we want to do is simply factor the denominator. This is extremely important. We can take out X, which leaves us with X multiplied by X + 1. And now we're going to attempt the direct substitution. We're going to get -1 + 5, because this is a number that is not 0. We're not worried about -1 from the left. But for the denominator, we're going to say that we get -1. This is not 0, so we can just put -1. And then for the other factor, this is where it becomes important. We get -1 from the left plus 1. Now, let's simplify this expression as much as possible. In the in the numerator, we get 4, but since we're dividing by -1, we can simply say that we end up with -4. And now for that factor where we have -1 from the left plus 1. We have to understand that if x approaches -1 from the left, this means that X is less than -1, and if X is less than -1, adding a value of 1 to it, well, this simply means that we're going to approach 0 from the left, right, because we have negative a number that is less than -1, so adding 1 to it will not exceed 0. So in the denominator we can say that we have 0 from the left. Now, let's understand that whenever we divide a negative number by a negative number, we get a positive number. So in this case, we get positive infinity. And now for the other limit, while we can essentially repeat all of those steps. But now we want to make sure that we're checking another limit. X approaches -1 from the right. We are factoring out the denominator. We're attempting direct substitution, right? And in this case we get -1 from the right plus 1 for one of our factors. Since we're approaching -1 from the right, this means that X is greater than -1. And if x is greater than -1, adding 1 to it gives us a value of 0 from the right. Now we are dividing a negative number by a positive number and a ratio of a positive and a negative number is a negative number. So we get negative infinity because we have a number divided by 0 with direct substitution. This by definition is infinity. And then we determine the sign based on the ratio of 2 numbers. If they're both negative numbers, just as in the previous case, we get positive infinity. If one of them is negative, the other one is positive, we get negative infinity. And notice that these are not equal to each other because they are not equal to each other, we can conclude that based on the limit existence theorem, the limit does not exist. Thank you for watching.

Determine the following limits.

c. lim x→−2 (x − 4) / x(x + 2)

Key Concepts

Limits

Factoring

Indeterminate Forms

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