2. Give counterexamples to show that these statements are false: (a) If f(c) is indeterminate, then the right- and left-hand limits as x → c are not equal. (b) If lim f(x) exists, then f(c) is not indeterminate. (c) If f(x) is undefined at x = c, then f (x) has an indeterminate form at x = c.

Q: If the fucntion f is undefined at x then lim f(x) does not exist. = c, C, FALSE O TRUE

A: Given- If the function fis undefined at x=c then limx→cfx does not exist. To Check- If the above…

Q: Use the Squeeze Theorem to evaluate the limit lim x→4 f(x)  if 8x − 16 ≤ f(x) ≤ x^2   on[2,6].

A:

Q: Show that lim (x,y) →(0,0) 16x4 +9y¹2 4x² + 3y6 does not exist.

A:

Q: Use the Squeeze Theorem to evaluate the given limit.  lim x→3 f(x), where 5x − 7 ≤ f(x) ≤ x^2

A: We know that Squeeze theorem states that: Ih h(x)≤f(x)≤g(x), when x is near a and…

Q: 4 Find the limit: lim t+0 t 4 et – 1, (Enter undefined if the limit does not exist.)

A:

Q: 2. Use the ɛ-d definition of the limit of a function to prove that lim = 3. I-»3 T – 2

A: Let  I  be an open interval containing  c , and let  f  be a function defined on  I , except…

Q: Let f(x)={5x+1  x≠4               {1       x=4. What is f(4)? If f(4) does not exist, type: 99999…

A: The given function is

Q: Use the definition of limits at infinity to prove the limit

A: Definition of limits helps to find the limit of the function. A value is said to be the limit of…

Q: Use the Squeeze Theorem to find limx→c f(x).

A: Given that 4-x2≤f(x)≤4+x2 To find the limit of the function using the Squeeze theorem when x→c=0.

Q: Determine the following limits analytically.

A: Given:  limx→1 2x-x2To find the Limits :Formula: limx→a f(x) =limx→a f(x)

Q: Let Зх — 2. if x + 2, f (x) = 5, if x = 2. (a) Find lim→2 f(x) (b) Prove your answer to Part (a)…

A:

Q: Let f(x) = 4x² + 6x + 2. According to the definition, f'(x) lim h→0 (Enter the limit in reduced…

A: Given f(x)=4x2+6x+2

Q: If f(x) is differentiable at X =c then lim f(x) must exist and f(c) must be defined

A:

Q: 4-Find the limits of the following x -1 1. lim Vx +3 - 2 Vx² +2 2. lim X+0 3x –6

A:

Q: Show that lim (x,y)→(0,0) 16x4+9y12 4x²+3y6 does not exist.

A:

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: 2. Use the definition of the appropriate limit to show (a) limr→∞(2x² + 4x – 1)/(3r³ + 1) 0. - (b)…

A:

Q: Let f(x) = 2x² + 2x + 3. According to the definition, f'(x) = lim h→0 (Enter the limit in reduced…

A: the soluton is following.

Q: lim 4e-4x lim 5ex - 4 X-4 9x + 7x - 7 lim 2x2 - 2x X- +0

A:

Q: Prove that if the limit of f(x) as x approaches c exists, then the limit must be unique.

A:

Q: Evaluate the following limits analytically, if it exists. (a) Vx+14 – 1 lim x→-10 2x (b) 5x +7x lim…

A: Given problem:-

Q: Evaluate the following limits (if exist): 1-cos(ax) (c) lim- a+0 (d) lim/Vxr'+x-Vx°-cx), where c is…

A: To evaluate the given limit we are using L- Hospital rule

Q: Use the E-8 definition of a limit to prove lim Xy (Xy) →(0,0) x² +y²

A: Consider the equation

Q: Show that the following limit does not exist. ху lim (x. y)-(0, 0) 3x2 +2y2

A:

Q: 4. Do you agree with the following statement? When a function f is continuous at a number a, then it…

A:

Q: Calculate the following limits x2+4x-12 а) lim x-2 b) lim Vx +1 In(x+ 1) |x-21 X→ー1

A:

Q: 1- Evaluate the following limits, if they exist. If they do not exist, state why. a- Lim 4 c- lim…

A: See the details solution in below

Q: 5. Show that x³y³ lim (x,y) →(0,0) 2x¹2 + 3y4 does not exist.

A:

Q: t2 - 1 lim t→-2 t2 +3t-1

A: Given limit is limt→-2t2-1t2+3t-1.

Q: Evaluate the limit, if it exists.

A:

Q: Use the e - 8 definition of the limit to establish a ô that proves lim (x – 1) = 2 x-2 O A) 3 B) 2e…

A:

Q: 10. Prove that the statement "If lim f(r) exists, then f is continuous at a" is false by using a…

A: See the solution below.

Q: If a function f is not defined at x = c, then limf(x) does not exist. True False

A: Limit of a function at a point exist mean left hand side limit and right hand side limit at that…

Q: 1. Show that the limit lim (2,y)(0,0) In(1 – a2)+ y* x² + y4 does not exist.

A:

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

A:

Q: Use the formal definition of the limit to prove the statement rigorously.

A: To prove: limx→0x2+x3=0. We know that limx→afx+gx=limx→afx+limx→agx Therefore,…

Q: 11. Show that the limit lim (z.y)→(0,0) r² + y4 does not exists.

A:

Q: Determine the following limits at infinity. lim x-11 xS-0

A: limx→-∞x-11

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

A:

Q: (2) Suppose that a function f(x) is defined for all real values x except x* x,, then lim f(x) does…

A: For limit to be exist at a point, left hand side and right hand side values of function should be…

Q: 44. Prove that if lim f(x)= L #0 and lim g(x)= 0, then the limit lim 8(x) does not exist.

A: Given: limx→cf(x)=L≠0 and limx→cg(x)=0 To Show- limx→cf(x)g(x) doesnot exist. Theorem- if limx→au(x)…

Q: A function f has a removable discontinuity at x = c if limf(x) exists by f is not continuous at x =…

A: Here, the given function is: f(x)=x-2|x|-2 So, function is not defined at x=2,-2

Q: Prove, using the formal definition, that lim,→1- V1– x = 0.

A: Given, limx→1-1-x=0

Q: 3. Compute the limit algebraically Vx +3x2 lim 6x 1

A:

Q: Evaluate the left hand and right hand limits of the function defined by (1+x, if 0 1 at x 1. Also…

A:

Q: Let F(x) = | 3e“ dt. Find lim F(x) if it exists. If the limit does not exist, explain why. x→0

A: see below the answer

Q: et –1 t2 – 3t + 5 lim t-0 t t2 – 1

A:

Q: 6. Explain why the limits lim f(x+h) – f(x) and lim f(2) – f(x) h Z-X are each initially in the form…

A: Given that the limits are limh→0fx+h-fxh and limz→xfz-fxz-x.

Q: 3. Compute the limit algebraically Vx + 3x2 lim 6x 1

A: Solution: The objective is to find the limit of the given function

Q: Suppose that lim, →o(f(x) – 9(x)) = 0 and h(x) is a function with domain R. a) Assume that |h(x) –…

A: We will answer the first question as we only answer one question at a time. Please resubmit the…