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 1/x−1/4 / x − 4

Accepted Answer

Welcome back, everyone determine the following limit or specify that it does not exist limit of one divided by Y minus one divided by three, divided by Y minus three. And Y approaches three, there are four answer choices. A negative 1/9 B one third, C 1/9 and D the limit does not exist. So we can begin solving this limit by direct substitution. Our function is one divided by Y minus one third divided by Y minus three. When we apply direct substitution, we get one third minus one third divided by three minus three. And this gives us zero divided by zero, which is an indeterminate form. So we have to eliminate that in determinate form. And we can see that by first of all, finding the common denominator for our numerator. So we get limit when Y approaches three. Now, for the first fraction, we noticed that we need to multiply both sides by three. For the second fraction, we need to multiply both sides by Y. The common denominator is going to be three Y and then we get three minus Y in the numerator. In the denominator, we have Y minus three. So Now, if we rearrange this into a single fraction, since we have double division, we get limit when Y approaches three of the numerator three minus Y divided by the product of the two denominators three, Y multiplied by Y minus three. Now we minus Y and Y minus three, those look quite similar. So we can actually factor out the negative sign or the numerator and this gives us negative Y minus three. So we are flipping the signs which gives us the same factor, right? Three Y multiplied by Y minus three. And we can clearly see that we can cancel out A Y minus we which gives us limit when Y approaches three of negative one divided by three Y. And now let's apply direct substitution once again, this is negative one divided by three multiplied by three, which is negative one line. This is a finite number, meaning we have successfully evaluated the limit and that's answer choice. A negative 19. Thank you for watching.

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Find the following limits or state that they do not exist. Assume a, b , c, and k are fixed real numbers.
lim x→4 1/x−1/4 / x − 4