20) Is the function given by f(x) ={x² - 4, for x <0 continuous at x = -2? Why or why not? for x 20 20) 1-2, A) Yes, lim f(x) = f(-2) x--2 B) No, lim f(x) = f(-2) does not exist x--2

Q: 35. Use the Squeeze Theorem to show that lim,-o (x² cos 20TTX) = 0. Illustrate by graphing the…

A: To show that: limx→0x2cos20πx=0 Using Squeeze Theorem:…

Q: f has domain of (-∞,–1)U(-1,2)U(2,00) lim f(x)=- %3D x→ -1" lim f(x)=- x→ -1* lim f(x)=3 x→2

A:

Q: Determine the limit: 12 x+ 1 tan(2 x) lim 3 sin(3 x) X- 0

A: Given limit: limx→012x+1tan2x3sin3x

Q: Fill in each blank so that the resulting statement is true.

A: We know, A function f is continuous at a when three conditions are satisfied. f is defined at a,so…

Q: 2. Explain why f(x, y) must approach a unique number L as (x, y) approaches (a, b) along all paths…

A: A number L is said to be the limit of a function f(x,y) as (x,y)→a,b , if for any real number ∈&gt;0…

Q: What is lim x-inf (x_n) and y=lim sup (x_n) where x_n = [(-1)^{3n}] + 2? (express answer as x,y)

A:

Q: Compute the derivative of f(x) = x³/2 using the limit definition. Hint: Show that f(x + h) – f(x) _…

A: We  will compute the derivative of given function  by using first principle of derivative

Q: Estimate the limit numerically or state that the limit doesn't exist lim x→0  sin(4x)/x

A: Before we get into the evaluation of this limit, let's recall the fundamental rule of limit as shown…

Q: 2. Show function f(x,y) = does not have limit of a function at point (0,0). 2² + ?

A: We need to show that the function does not have limit at point (0,0)

Q: 5. Find the limit or show that it does not exist. sin(x – 1) lim 2 x→1 x´ + x -2

A:

Q: lim as x--&gt;0  of (x+1-cosx)/x

A: The given limit of the function:

Q: Find the limit (if it exists) (а) lim x — 5 (b) lim 5 x-243 x2 + 6x + 8 (c) lim x--2 x + 2 sin x (d)…

A:

Q: Find the limits by using L.Hopital rule : 7x 1. lim x-0 sin 2x x2-16 2. lim x4 (Vx+5)-3

A: We need to find the limits of the given expressions by using the L.Hopital rule. For (1),

Q: 1. Find the derivative of f(x) = 2. at x = 0 using the definition of the derivative as a limit. Va +…

A:

Q: 5. If lim f(x, y) = 5 (x,y)→(3,–7)" %3D for every curve C in R? containing the point (3, -7), what…

A: If limx,y→3,−7fx,y=5  is true, then it represents that limit exists atpoint (3,-7) for function…

Q: limit as x----0 for the function (px)/(sqrt(bx+a^2))-a using L'Hopitals

A: Apply the limit.

Q: sin(x – 3) lim X→3 + 6x – 27

A:

Q: limit as x---3 for the function (x-2)^11/(3-x) using L'Hopitals.

A: Consider the given function (x-2)11 / (3-x).

Q: x+4 Which will prove that f(x) = %3D x2+16 is continuous at C= -4? 1. f(-4)=0 II. lim f(x)=0 III.…

A:

Q: Prove that lim x-&gt;0+ √ x [1 + sin2 (2 π/x)] =0

A: Given: limx→0+x1+sin2 2πx By using the property of limit, the limit of a product is the product of a…

Q: Use the Squeeze Theorem to show that lim (x2 sin(2)) = 0 x-0 %3D

A:

Q: 7. If a continuous, differentiable function f has zeroes at x = -4, x = 1, and x = 2, what can you…

A: Mean value theorem relates average rate of change of a function over an interval to its…

Q: Find local maximum or local minimum of f(x) = - sin 2x -x defined on 2'2

A:

Q: 2 Find the derivative of f(x) =x² – 6 from limit definition. - Show all steps of working. f(x) :…

A:

Q: 3. Find the limit. tan(5x) 0 sin(3x) sin(6x) (2) lim x04x-5x (1) lim

A:

Q: Limit as x approaches 2 of (x3-8)/(x-2)

A: Obtain the value of the limit as follows.

Q: A function is defined by: In( x- 1) f (x) =- 1-x a) Using L'Hopital's rule, show that: lim f (x) =0,…

A: Critical points

Q: 6. State the definition of the derivative of a function f. Use the definition of the derivative to…

A:

Q: In(1+ x?) lim x-0 x3

A:

Q: Fird the domain of f (x) = 2-Vx2-5 Solve lim x→-3 =? (Don't use L'hopital Rule) x+3

A:

Q: How would you define f(2) in order to make f continuous at 2? limx→2 x^3-8/x^2-4 show ALL of your…

A: Let's solve limit problem.

Q: compute limit as x goes to infinity of xe^-x

A: Given,            limx→∞x e-x

Q: 9. Is the function defined by gon3 – 3 if 2. continuous at r= 2? differentiable at x = 2?

A:

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: 5. Find the limit or show that it does not exist. sin(x – 1) lim x→1 x´ +x –2 2

A:

Q: Suppose f is continuous on the interval (0,3). Assume further that f(0)= -2 and f(-3)= 5. Also, lim…

A:

Q: Find local maximum or local minimum of f (x)=sin 2x-x defined on 2 2

A: The given function is fx=sin 2x-x. Obtain the critical points as follows. Find f'x as…

Q: 9 -x2 if x 1 .What value of c makes g(x) continuous at x = 1? Show all conclusions. 2 Vx-4 Using…

A: For a function to be continuos, the left hand limit should be equal to the right hand limit.…

Q: 4. a) Explain using theorems from Section 2.5 why the function is continuous at every number in its…

A:

Q: 4) Use the Limit Definition of a derivative to find f '(3) if f(x): %3D 5-x

A: Given:- f(x) =2/(5-x)  To find:- Use limit Definition of a derivative to find f'(3)

Q: . Suppose f(т/3) — 5, f (т/3) — —3,а sin(3та) 2->0 sin(5Tx) . Find the limit: lim

A:

Q: a) f(a) exists b) f(x) is defined for 0a f(x) exists e)lim x-->a f(x)=f(a) which ones are true?…

A:

Q: Q3 :(A): By using "L' Hopital rules "; Find the limits of : (2) limx-o x-³ (tan-¹x - x)

A:

Q: b) By using "L'Hopital rule"; Find the limits of:- (1) lim,o In (sec x) 3x 2) lim,* (tan x)*

A:

Q: 2) Use the definition of the derivative f(x+h)-f(x) lim hー→0 to find the derivative of f(x) =…

A: Our objective is to find the derivative using definition.

Q: lim x→0 for sin(4x)/3x

A:

Q: 2. Evaluate the limit by Use the definition of the derviative to compute the derivative of f(x) = 1…

A: Given,

Q: x-3- – 3VA lim x+0* 0" x- Vx

A: We first try to find the limit  by substitution