Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Evaluate limit and justify your answer.$$\lim _{x \rightarrow 0}\left(\frac{x}{\sqrt{16 x+1}-1}\right)^{1 / 3}$$

Calculus 1 / AB

Chapter 2

Limits

Section 6

Continuity

Harvey Mudd College

University of Michigan - Ann Arbor

University of Nottingham

Boston College

Lectures

03:09

In mathematics, precalculu…

31:55

In mathematics, a function…

04:03

Evaluate $\lim _{x \righta…

01:41

Evaluate limit and justify…

02:26

Determine the following li…

Evaluate the limits using …

02:47

evaluate the limit.$$<…

00:45

Find the limits.\begin…

00:42

Find the limits.$$\lim…

02:21

00:55

01:06

Find the indicated limits.…

I want to find the limit as X goes to zero of the function X divided by Route 16 X plus one minus one all to the power of 1/3. Um, so first will just use the limit rule for roots up. Teoh Rewrite. This adds parentheses limit as X goes to Ciro X Group 16 X plus one. But it's one, and we're raising the entire limit to the 1/3 power. So that allows us to sort of disregard that, uh, 1/3 exponents until after we brought you the inner limit. Uh, so to do that, since we have Well, first we want toe check. Direct substitution. We can see that if we plug in zero into the denominator, we're gonna get route 16 times their A plus one minus one. Uh, which is route one minus one, which is zero. So direct substitution is not going to work. Um, you to try something else? So since we have a radical, uh, minus one, we're going Teoh, let's multiply at the top of the bottom by the contra kit. So that's going to give us get the parentheses going, uh, limit as X goes to zero of X times and then the contra git is route 16 x plus one plus one changed sign divided by, um, and then multiplying that that, uh, the same expression by the original denominator, the radicals gonna cancel out. So we'll get 16 x plus one, and then the minus one times plus one will give us minus one so the ones will cancel out. So the denominator is just 16 X. Uh, so this expression is still equal to our original limit. And now this is a zero, Um, and so we can see that we tried direct substitution here plugging zero and gives us zero in the denominator. So that's not gonna work. Um, but we do have an X, and he has a factor of the numerator and the denominator. So if we just look at the left or right limit by itself. So let's say we just look at the limit as X goes to zero plus well, on the right, X is not equal to zero. Which means that if we wanted to find the right limit of this expression, we could weaken, cancel out the exes from the numerator and the denominator and get six, Route 16 x plus one plus one divided by 16. Um, and so this is this no longer has a serum denominator because he has 16 so we'll just, uh, used directs. Institution plug zero into X 16 times zero plus one plus one worker 16. Um, and that gives us to over 16 war 1/8. Um, so we can see that if we were to take clear to change this to the limit as excuse to zero minus, we could apply the exact same calculations. Excellent. Cancel out, because X isn't zero to the left of zero. Um, and then we would use direct substitution. So the limit as X goes to zero minus is also 18 since the left and right limits are equal. Um, we can conclude that the original limit as X goes to zero up here is also equal to one. It just by definition, um, so, uh, continuing from up here, we can plug in 1/8 for the limit, and it's still raised to a 1/3 power, and that gives us 1/2. So our limit is excuse to zero is just 1/2

View More Answers From This Book

Find Another Textbook

In mathematics, precalculus is the study of functions (as opposed to calculu…

In mathematics, a function (or map) f from a set X to a set Y is a rule whic…

Evaluate $\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{x-1} .$ (Hint: $x-1=(…

Evaluate limit and justify your answer.$$\lim _{x \rightarrow \infty}\le…

Determine the following limits.$$\lim _{x \rightarrow-\infty} \frac{\sqr…

Evaluate the limits using the limit properties.$$\lim _{x \rightarrow 16…

evaluate the limit.$$\lim _{x \rightarrow 1} \frac{\sqrt{8+x}-3 x^{1…

Find the limits.\begin{equation}\lim _{x \rightarrow 1^{+}} \sqrt{\frac{…

Find the limits.$$\lim _{x \rightarrow 1^{+}} \sqrt{\frac{x-1}{x+2}}$$

Determine the following limits.$$\lim _{x \rightarrow-\infty}\left(-3 x^…

Find the indicated limits.$$\lim _{x \rightarrow 0^{+}}\left(\frac{1}{\s…

03:03

The cost, $C$ (in dollars), to produce $g$ gallons of a chemical can be expr…

02:02

Determine the points on the interval (0,5) at which the following functions …

03:10

Prove the following identities.$$\sin ^{-1} y+\sin ^{-1}(-y)=0$$

Determining limits analytically Determine the following limits.$$\li…

01:28

Find the relative rate of change, $f^{\prime}(t) / f(t),$ of the function $f…

01:49

Determining limits analytically Determine the following limits.$$\lim _{…

01:16

Describe the parallels between finding the instantaneous velocity of an obje…

04:34

Explain the relationships among the slope of a tangent line, the instantaneo…

01:11

01:09

What are the potential problems of using a graphing utility to estimate $\li…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.