Answered: Let f:[a,b] → ℝ, g:[a,b] → ℝ, and x ∈…

Let f:[a,b] → ℝ, g:[a,b] → ℝ, and x ∈ [a,b] such that f′(x) and g′(x) exist, g′(x) ≠ 0, and f(x) = g(x) = 0. Prove lim?→? f(t) / g(t) = f′(x) / g′(x)

College Algebra

1st Edition

ISBN:9781938168383

Author:Jay Abramson

Publisher:Jay Abramson

Chapter3: Functions

Section3.3: Rates Of Change And Behavior Of Graphs

Problem 2SE: If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local...

See similar textbooks

Related questions

Q: Consider the limit limx→0 1 x 2 = ∞ and let M = 1000. Find the largest value of δ > 0 such that if x…

A: The definition of limit: Let fx be a function and limx→cf(x)=∞. Then, for every M>0, there exists…

Q: Compute the limit, or show that it does not exist: lim (x,y)→(0,0) x³y? + x²y³ x4 + y4

A: To Determine :- Compute the limit or show that it does not exist : lim x , y → 0 , 0 x3 y2 + x2…

Q: Let f be defined by: 1, x rational f(x)={ 0, x irrational…

A: To prove that for no number C foes limit x tends to c f(x) exist.

Q: Prove, using the e-d definition of limit, that: (a) lim x² + 1 = 2 24-1

A: As per our guidelines, we can answer only one question in a single request. Kindly re-post the other…

Q: Find the limit. lim x→0+ (- 7lm a)" In x

Q: In (x +h) – In (x) is When x = 2e, lim h→0 h

A: Substitute x= 2e above

Q: Consider the function f(x) = vx for xE R and 0 <x < 32. Prove that lim,-s f(x) = 4. %3D

A: Given: f(x)=x for x∈ℝ and 0<x<32 we have to prove that limx→16f(x)=4

Q: Prove that lim r(x)= where m = 0, 1, 2, 3. .. II

Q: Plot the graphs of u(x) = 1+ |x – and I(x) = sin x on the same set of axes. What can you say about…

A: Sandwich theorem of limit states that if our original function is sandwiched between two functions…

Q: 1- Evaluate the limit a-Lim(x/(x-1) – 1/ Inx) b- lim [In(cos(x))]* c- 2-VX²+4 /sin'x x->1+ x >0 x>0…

Q: and If lim f(x) = lim g(x) = +o and flx) =1 and gtxl = e, what is the limit of the division of !!…

A: As given limit is infinitely by infinity form so use L'hopital's rule and substitute the given…

Q: Suppose fé Ra, b]. Prove that b- a lim b- f(x) dr. k=1

Q: Prove that lim (x,y)(0,0) x² + y² does not exist by considering the limit along the x-axis.

A: Given: lim(x, y)→(0,0)xx2+y2

Q: Let f : (a, b) → R be differentiable and let c ∈ (a, b). Suppose that limx→c f‘(x) exists and is…

A: handwriting solution

Q: Prove that if lim, „a g(x) = 0 and lim, „«f(x) exists and is not 0, then %3D f(x) lim does not exist…

Q: Evaluate f(-1+h) – f(-1) lim where f(r) = 5z'+1. Enter I for oo, -I for -oo, and DNE if the limit…

Q: lim (sin?x x-0 x2

A: We use L-Hospital Rule to evaluate this limit.

Q: Let f be defined in (a, b) and differentiable at xo E (a, b). Let (xn), (Yn) be such that a < xn <…

A: Here we need to construct a bounded sequence {λn} , which is bounded.

Q: Evaluate the limit lim z→2+i [(z−1)(3−2i)+1−3i] and use the definition of the limit (−δ) to…

Q: Prove that lim(x,y)→(1,2) (x2 + 2y) = 5 by using the precise definition

A: ε-δ definition: Limit of function f(x,y) at a point (a,b) is 'L' if for every ε>0 there exist…

Q: Suppose that f : R -R is differentiable at z = 0. (a) Prove that (t) lim,-o EA = f'(0) (b) Give an…

Q: Evaluate the limit if it exists, or show that it doesn't exist: xyz lim (a7,1,2)¬(0,0,0) [ æ² + y² +…

A: Evaluate the limit lim(x,y,z)→(0,0,0)xyzx2+y2+z2=0·0·002·02·02=00

Q: find all values of c such that the limit exists

A: Given

Q: Prove that if lim x(t) = X and lim y(t) = Y, then lim(x(t), y(t)) = (X, Y). t-a t-a t-a

A: Now, xt, yt=∫cdxtytdt Given: limt→axt=X , limt→ayt=Y

Q: suppose f,g an d h are functions which g(x)<=f(x) <=h(x) ,for all x in an open interval containing a…

A: No. For the given continuous open interval domain, the limit of f(x) does not exist.

Q: Find limit, or show that it does not exist. 2x²y (x,y)¬(0,0) x*+y2 lim

A: Consider the given: limx, y→0, 02x2yx4+y2Find two different paths to approach the point that gives…

Q: If A(t) is continuous at m, then what is the value of lim A(t) ? t-m If h(f) is continuous at G,…

A: If A(t) is continuous at m, then limit t goes to m should be equal to value of function A at m i.e…

Q: 2. Use Squeeze Theorem to find lim,x f(z), if for all z >4 6e - 15 < {(x) < VI- 2 2e

Q: Prove, using the formal definition, that lim→-1 (T+1)² +0o. %3D

A: We have to find the limit limx→-13x+12=+∞.

Q: Let f(x) = 8x + 3. (a) Prove that lim f(x) = 27 using the formal definition of the limit. 3 (b) Find…

A: Givenf(x)=8x+3To prove limx→3f(x)Subdivision b is not defined properly

Q: use the definition of differentiability f'(x) = lim- h-0 f(x+h)-f(x) h to For f(x) = %3D 6. (x-1)…

A: fx=x2x-1 Now,…

Q: Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths in…

A: We have to explain why fx,y must approach a unique L as x,y approachesa,b along all paths in the…

Q: Prove, using the e-d definition of limit, that: (a) lim x² + 1 = 2

Q: lim cos x*/x x→0

A: Given, limx→0cosx1x

Q: Using the squeeze theorem, find the value of lim,0 x² sin=) O 1 O -1 O does not exist

Q: Shone that lim g (x,yJ-p (0,0) does not exist,

A: According to the definition of limit, we have

Q: Evaluate using the Squeeze Theorem.

A: Given: limx→0 x sin 1x2 To evaluate: The limit using Squeeze theorem.

Q: Use the Squeeze Theorem to show that lim (x² cos 21xx) = 0. Illustrate by graphing the functions…

Q: Let zo = 0. Discuss if the limit of function f(2) = exists at zo- • If yes, then lim f(z) = z+z0 •…

Q: 58. If f(xo, Yo) 3, what can you say about lim f(x, y) (x, y)→(xo, Yo) if f is continuous at (xo,…

Q: Show that the lim Z→0 |Z| does not exist.

Q: Use the Squeeze Theorem to show that lim x* cos = 0. If If(x) < B| for all x in an interval…

A: Squeeze (Sandwich)Theorem The Sandwich Theorem, also known as the squeeze theorem, is used to find…

Q: Consider the function f (x) = x(Va – 1), where a > 0, x > 0. If 0 < a < 1, show that limx0f (x) is a…

Q: Evaluate the limit below. 1 lim (x,y)→(0,0) 1-x2-y?

A: limx, y→0, 011-x2-y2

Q: 1. Show that the limit does not exist. xy? lim (x,y)→(0,0) x² + y4

A: This question is related to limit in two variables.

Q: Evalvate the limit lim Cos x In(x- TT Xーつク nピ-e)

Q: Find lim 4(x) (x) / X-0 x 2. Prove that B(m, n)

A: Limit of the bessel function

Q: Find Limit Using Squeeze Theorem: 3. If 2x s g(x) s x* – x² + 2 for all x, find lim g(x).

Q: Prove that lim n 00 a" f (x) dx = 0, for any f E R[0, 1].

Q: What is the the value of the limit lim,0 Vn (Vn+1- /n)? Show your work.

Question

Let f:[a,b] → ℝ, g:[a,b] → ℝ, and x ∈ [a,b] such that f′(x) and g′(x) exist, g′(x) ≠ 0,
and f(x) = g(x) = 0. Prove lim_?→? f(t) / g(t) = f′(x) / g′(x)

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

VIEW

Step 2

VIEW

Trending now

This is a popular solution!

Step by step

Solved in 2 steps with 2 images

SEE SOLUTION Check out a sample Q&A here

Similar questions

If a functionfis increasing on (a,b) and decreasing on (b,c) , then what can be said about the local extremum offon (a,c) ?
The limit of the function f as x approaches a is the number L, written as limx→af(x) = L, if for every ε > 0, there exists δ > 0, such that if 0 < | x - a | < δ, then | f(x) - L | < ε. Using this definition, prove that limx→1+ 4 / (1-x) = -∞.
Find formulas for f and g such that limx→∞f (x)/g(x) is an ∞/∞ form that has a limit of ∞.
Prove rigorously that lim lxl = 0.x→ 0
Consider g(x)=−7x3+7x2+9. a) limx→0−g(x)= b)limx→−1−g(x)= c)On what interval(s) is g(x) continuous?
Suppose that functions g(t) and h(t) are defined for all values of t and g(0) = h(0) = 0. Can limt-->0 (g(t))/(h(t)) exist? If it does exist, must it equal zero? Give reasons for your answers
Consider the function f(x) = 1/x for x ∈ ℝ and 3 < x < 7. Prove that limx→5 f(x) = 1/5. using epsilon and delta method
Consider the function f (x) given by f(x) = {kx, if x < 1; 0 if x = 1; 2k-kx if x > 1}, where k > 0 is a constant.a) For each of the three conditions for continuity, determine whether it is satisfied at point x = 1.b) Based on a), is f(x) continuous at x = 1? Does limx->1f(x) exsist? Is limitx->1f(x)=f(1)?
Suppose X⊆ R, a∈ X', f: X -> R, lim {x ->a} f(x) = L, and c ∈ R, is a constant. Use theorems to prove lim {x-> a} cf(x) = cL.
suppose f,g an d h are functions which g(x)<=f(x) <=h(x) ,for all x in an open interval containing a , except possibly at a. if lim g(x) ,x approaches to a does not exist or lim h(x) ,x approaches to a does not exist , will the lim f(x) ,x approaches to a do or does not exist ?
Consider the limit limx→0 1 x 2 = ∞ and let M = 1000. Find the largest value of δ > 0 such that if x ∈ (c − δ, c) ∪ (c, c + δ), then f(x) ∈ (1000,∞). That is, now close does x have to be to 0 so that 1/x2 is in the interval (1000,∞).
Prove that if the limit of f(x) as x approaches c exists, then the limit must be unique. ( Hint: Let lim f(x) = L1 as x approaches x and Lim f(x) = L2 as x approaches c and prove that L1=L2.