Find the following limits or state that they do not exist. Ass...

Question

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.

lim x→4 3(x − 4)√x + 5 / 3 − √x + 5

Accepted Answer

Welcome back, everyone evaluate the following limit or state. If it does not exist limit, when T approaches nine of four, T minus nine, multiplied by square root of T plus seven, divided by four minus square root of T plus seven. There are four answer choices. A negative 64 B negative 128 C zero. Indeed, the limit does not exist. We can begin solving this problem by direct substitution. T approaches nine. In the numerator, we have four multiplied by T minus nine, multiplied by square root of T plus seven divided by four minus square root of C plus seven. When we apply direct substitution, T minus nine becomes nine minus nine, which is zero. And the whole product is going to be zero. In the denominator, we have four minus square root of nine plus seven. So four minus square root of 16 which is four minus four. And that's zero, right? So zero over zero, divided by zero is an indeterminate form. So if this is an indeterminate form, our goal is to eliminate it to see if there's a possible hole in this rational function. And we can see that by multiplying both sides of our function by the conjugate of the denominator. So instead of having a negative sign, we're going to replace it with a positive sign between four and the radical term. So we end up with limit. When C approaches nine, let's begin with the denominator four minus square root of T plus seven. We want to multiply that by the conjugate, which is four multi four plus square root of T plus seven. And we want to multiply the numerator by the same term. And that's four plus square root of T plus seven. Now, for the denominator, we can see that this is the factor form of the difference of squares. If we go in reverse, we can notice that this gives us a squared minus B squared because A squared minus B squared is equal to A minus B multiplied by A plus B. And this is why we multiplied by the conjugate in the first place to essentially go in reverse and evaluate the denominator as follows. While our A values for so for square, the 16, our B term is the radical term. So we are subtracting the square of square root of T plus seven which is just T plus sub. Now, for the numerator nothing changes. We're just going to leave it as it is for. Now, we have four T minus nine square root of C plus seven multiplied by four plus square root of T plus seven. We don't want to multiply anything just yet. We are just going to ma we are just going to simplify the denominator, starting with the denominator. We have 16 minus seven minus T. So that's nine minus T and we have something similar in the numerator that's T minus nine. So what if we factor out the negative sign and then flip the signs for the denominator terms? This gives us T minus nine with the negative sign in front, right? Because we can distribute it and get nine minus T. And this essentially tells us that we are now able to cancel out T minus nine for each side. Let's go ahead and do that. This essentially eliminates the whole. And we can try and apply direct substitution once again, we can factor out the negative signs. We get negative limit. When Q approaches nine in the numerator, we have four square root of T plus seven multiplied by four plus square root of T plus seven, there is no denominator negative one. We have our negative sign in front. So we can now apply the direct substitution which gives us four multiplied by square root of nine plus seven. That's 16. So square root of 16 is four, we get four multiplied by four, multiplied by four plus square root of 16 witches four plus four. And now this can be simplified to 16 multiplied by eight. And we have our negative sign and fronts, we get negative 16 multiplied by eight, which is equal to negative 128. This is a finite number, meaning we have evaluated the limit and the answer choice is B negative 128. Thank you for watching.

Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→4 3(x − 4)√x + 5 / 3 − √x + 5

Key Concepts

Limits

Rational Functions

L'Hôpital's Rule

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