Fill in each blank so that lim f(x) exists x→2 1 Ex+1, x < 2, _, x < 2, f(x) =- x + f(x) = {2 3x +_ r2 2. -x +6, x > 2. x² -5.x+6 x² – 9, x< 2, x<2, f(x)=- f(x)={ x- 2 -x + r2 2. x>2.

Q: 2) Show that the_lim. 1) xy2 lim (x,y)→(0,0) x² +y²

A:

Q: 2. Construct a table of values to investigate 1 if x -1

A:

Q: 4-If. 2-x < flx) < 2Cos(x) for all x then find: lim f(x)

A:

Q: Evaluate the limit. 8 lim x→4+ \x-1 x + 2, / \4 – x/

A: To evaluate the limit: limx→4+xx-1-8x+214-x Solution: First we will simplify the expression and then…

Q: 5. a) Explain why lim vx-3 does not exist. b) What about lim V(x 3)2 ?

A:

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: tan 3h 3. Estimate the value of lim,0- by creating a table of function values for h h = 0.01,0.001,…

A: We need to substitute these values of h in function f(h) to evaluate limit

Q: Prove that lim, = 1, (x > 0).

A:

Q: B. Find the limit. Зх — 1 5. lim х-1/3 9х2-1 x3 - 1 6. lim х+1 х-1

A: As per our guidelines we are supposed to answer only 3 subparts

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: Let f(x) = 3x + 2. Use the € – d definition to prove that lim 3x +2 1. x→1 Definition. E-d proof.

A:

Q: Evaluate the limit. 1-x² lim *+1 sin(27x) a) lim b) x+2x² C) lim - - 2x *→(π/2)- (2-) tan r Show all…

A:

Q: find the limit or explain why it does not exist.

A: To find the limit  limx→π sec x = ?

Q: Evaluate the limit or determine that it does not exist.

A: Given that limx, y→0, 0 tanx2+y2 tan-11x2+y2. Compute the limit as follows.

Q: 2 lim x→1- x²

A: The limit of the function f(x) is obtained as L=limx→a-f(x)L'=limx→a+f(x) Then the limit of the…

Q: 7. Find the limit, if it exists. If the limit does not exist, explain why. - (a) lim sin (2x 6)…

A: solution is given below

Q: Let F(x) = (x2 + 3x + 2)/(2 - | x | ). Make tables of values of F at values of x that approach c =…

A: Given: Fx=x2+3x+22-x To determine: We have to estimate limx→-2Fx by making a table of values of F at…

Q: Find the limit (if it exists). If it does not exist, explain why.

A:

Q: lim -1 x→0+

A: Separate the limit.

Q: Find a number b such that lim- .2. x²-4 =2 x-b x-x -2

A:

Q: 1. Explain what lim f(x, y) = L means. (x, y)-(a, b)"

A: see below the explanation

Q: Given that lim f(x) X+ 0 x2 1- explain why lim f(x)= ∞, x2 +7f(x)- sin (5x)+ 1 2- evaluate the limit…

A: We have to find the following

Q: Sketch the graph of the function and use it to determine the values of a for which limx--a f (x)…

A: We will find out the required values.

Q: 4-If. 2-x < f(x) < 2Cos(x) for all x then find: lim f(x)

A: I applied sandwich theorem which is defined as

Q: V2x –1 Using defination of limits, prove that lim Vx- 3 L^ = X4

A:

Q: Find a formula for a function f that satisfies the following conditions. lim fx) = 0, lim f(x) - -0,…

A: Function

Q: 5. If f(a) is undefined, then the lim S(x)does not exist. True or False? Explain.

A:

Q: Using the ɛ, & definition, prove that lim V = 1. %3D x 1

A: We have to prove the given limit.

Q: Prove that limx-&gt;∞ f(x) =lim t-&gt;0+ f(1/t) and  limx-&gt;∞ f(x) =lim t-&gt;0- f(1/t)   if these…

A: We have to prove limx→∞fx=limt→0+f1t Take limx→∞fx Put t=1x when x→∞ then t→0 we get…

Q: 4. If lim f(x) = -2, lim f(x) = 3, and f(0) = -2, what can X0+ X0- you say about lim f(x)?

A: Given, limx→0+f(x)=-2limx→0-f(x)=3f(0)=-2     Find limx→0f(x)

Q: Find the limit. lim (2 – x)3 tan(a) x→1

A: limit

Q: lim (x³ + 4x² + 4x – 1) = -1, a-2 1 lim - = 1.

A: Given,

Q: Show that 2 x² - y² - 1 lim (x,y) →(1,0) 1 − x + (2y)² - does not exist.

A:

Q: Use continuity to evaluate the limit. 16 + Vx lim 16 + x

A:

Q: ах Determine the values of a and b such that lim Vax +b – 3 = 5. x→0 bx

A:

Q: Determine the infinite limit. lim 3 sec(x) x- (T/2)+ X 00 - 00

A: Given limx→π2+3xsecx, to find the limit

Q: Find lim,-2 f(x) 2 -3 -2 -1 2 3 lim→-2 f(x) = 0 %3D lim, -2 f(x) = 2 lim,-2 f(x) = 1 Does Not exist

A:

Q: Calculate the limit.

A: limx→∞tan-11+x1-x

Q: -x - 4, x 3

A:

Q: Evaluate the limits: lim x tan X- а. x→1+ b. lim x1/x X→0+

A: a)  Applying the product rule, writing limx→1+xtanπx2 as…

Q: show that lim(x,y)→(0,0) (x^3+xy^3/x^4+y^2) does not exist.

A: We will see that along y=mx2 path limit does not exist which means limit does not exist.

Q: ). Determine the interval(s) on which f(x) =4x−12x−1is continuous. b). Find lim x→0…

A:

Q: State whether lim f (x) and f(c) are equal or unequal at the given value of c. Also, state whether…

A: Limiting value of the function is value that function seem to approach as it reaches certain value.…

Q: Find the limits limx→0+x tan (π/2 - x)

A: Given that, f(x)=x tan (π/2 - x) We have to find, limx→0+xtanπ2-x

Q: Prove that lim x-y does not exist. (x.y)(0.0) x + y

A: lim(x,y)-(0,0) (x-y)/(x+y) does not exist.

Q: Use continuity to evaluate the limit. 12 + Vx lim x-4 V12 + x

A:

Q: prove that lim x-2 f(x)=4 if f(x)= x^2, x does not =2…

A: Given,           To prove limx→2f(x) = 4 and f(x) =x2      if  x≠21        if x =2

Q: lim(x,y)->(0,0)  (4x2y)/(x2+y2)1/2

A: Given: lim(x,y)→(0,0)4x2yx2+y2         (1) To compute: The limit.

Q: Use e-5 definition and prove that lim (2 5r) -6

A: Here we use basic definition of epsilon and delta .

Q: Evaluate limx=0 (e-ax - e-bx)/x

A:

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.