1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Mademuasel [1]
3 years ago
14

CALC- limits please show your method

Mathematics
1 answer:
gladu [14]3 years ago
3 0
A. Factor the numerator as a difference of squares:

\displaystyle\lim_{x\to9}\frac{x-9}{\sqrt x-3}=\lim_{x\to9}\frac{(\sqrt x-3)(\sqrt x+3)}{\sqrt x-3}=\lim_{x\to9}(\sqrt x+3)=6

c. As x\to\infty, the contribution of the terms of degree less than 2 becomes negligible, which means we can write

\displaystyle\lim_{x\to\infty}\frac{4x^2-4x-8}{x^2-9}=\lim_{x\to\infty}\frac{4x^2}{x^2}=\lim_{x\to\infty}4=4

e. Let's first rewrite the root terms with rational exponents:

\displaystyle\lim_{x\to1}\frac{\sqrt[3]x-x}{\sqrt x-x}=\lim_{x\to1}\frac{x^{1/3}-x}{x^{1/2}-x}

Next we rationalize the numerator and denominator. We do so by recalling

(a-b)(a+b)=a^2-b^2
(a-b)(a^2+ab+b^2)=a^3-b^3

In particular,

(x^{1/3}-x)(x^{2/3}+x^{4/3}+x^2)=x-x^3
(x^{1/2}-x)(x^{1/2}+x)=x-x^2

so we have

\displaystyle\lim_{x\to1}\frac{x^{1/3}-x}{x^{1/2}-x}\cdot\frac{x^{2/3}+x^{4/3}+x^2}{x^{2/3}+x^{4/3}+x^2}\cdot\frac{x^{1/2}+x}{x^{1/2}+x}=\lim_{x\to1}\frac{x-x^3}{x-x^2}\cdot\frac{x^{1/2}+x}{x^{2/3}+x^{4/3}+x^2}

For x\neq0 and x\neq1, we can simplify the first term:

\dfrac{x-x^3}{x-x^2}=\dfrac{x(1-x^2)}{x(1-x)}=\dfrac{x(1-x)(1+x)}{x(1-x)}=1+x

So our limit becomes

\displaystyle\lim_{x\to1}\frac{(1+x)(x^{1/2}+x)}{x^{2/3}+x^{4/3}+x^2}=\frac{(1+1)(1+1)}{1+1+1}=\frac43
You might be interested in
The difference of x and y is 14. The value of x is 3 more than twice the value of y. Write two equations and solve/graph to find
vazorg [7]

Answer:

b

Step-by-step explanation:

4 0
3 years ago
Point M is located at (7,1).
Charra [1.4K]

I think it is possibly point b

8 0
2 years ago
A triangle with point (8, 10) undergoes a dilation with a scale factor of 1/2. What are the new coordinates of the point?
Lorico [155]

Answer:

(4, 5)

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
malia has $500 to purchase water bottles and pairs of socks for a fundraiser for her school's cross country team. She needs to b
BigorU [14]

Answer

6.9step-by-step explanation: simple math really6 1/2 times 0.6 = 3.9but if you are just doing 6

Step-by-step explanation:

4 0
2 years ago
Can you please help me with this
dem82 [27]

The answer is 63

Glad I could help :)

6 0
2 years ago
Read 2 more answers
Other questions:
  • how are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional situations?
    6·1 answer
  • Evaluate 3 - 24 + 8 + 42.<br> A.0<br> B.2<br> C.16<br> D.24
    13·2 answers
  • What is the volume of the cylinder?
    10·1 answer
  • Help me plis What is the exact measurement of the line segment? A. B. C. D.
    7·2 answers
  • Pls help! It has to be due by 5pm.
    11·2 answers
  • Write a <img src="https://tex.z-dn.net/?f=y%3D%5Cfrac%7B4%7D%7B5%7Dx-2" id="TexFormula1" title="y=\frac{4}{5}x-2" alt="y=\frac{4
    11·1 answer
  • Can anyone help me ??
    9·1 answer
  • Tyler further earns $12 per hour the store offers him a raise a 5% increase per hour after the race how much would Tyler‘s broth
    11·1 answer
  • Help meeeee!!!!!!!!!!! Also I have more of these sooo
    7·1 answer
  • Who ever gets it I will give Brainly and 20 points
    10·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!