Let ax², x < 1 f(x) = 1, x = 1 COS TX, x > 1 Find the value of a, such that lim f(x) exists and is finite. x 1

Q: Let f(x) = sin(sin(sin.x)) for for all x in R (f(x) x-0 g (x), and g(x) = (sinx), VxER, then lim is…

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Q: Suppose that f(x, 0) = 3 for all x ≠ 0 and f(0, y) = 5 for all y ≠ 0. What can you conclude about…

A: The limit of the function, f(x,y) at a point (a,b) does not exist if two paths upon which the…

Q: (4) Let f(x) be the function whose domain is all of R given by the rule 42 if r € Z and f(x) = 10…

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Q: Q8. We shall formally show that if two functions f and q are both differentiable at c, then we have…

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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…

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Q: If f(x+ y)= f(x)f (y) for all x, y eR and f (x)=1+g(x)G(x), where lim g (x)=0 and lim G (x)…

A: The problem has been solved in detail in the next step.

Q: If lim f(x) = = co then f (x) XCOS cos² X00

A: Given- If limx→∞fx=∞ then fx=xcos1x To Prove- The above expression.

Q: x² 1) Let f(x): = (Figure 24). Verify the following: x - 1 (a) f(0) is a local max and f(2) a local…

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Q: X,x1 Establish the existence of limf(x)· x-1

A: Given function is  fx=x,  x&lt;11,   x=1x2, x&gt;1

Q: Consider the function 1– cos(x) f(x): (a) (T) Evaluate lim, →0 f(x) = L. (b) (T) As x → 0, at what…

A: "Since you have posted a question with multiple sub-parts, we will solve first three sub-parts for…

Q: Let f(x) = Prove that lim f(x) = 0. x →0 sin x + sin 3x + sin 5x, if x is rational, 0, if x is…

A: We need to prove that limit is zero.

Q: Let f(x) = x²sin () for x + 0 and f (0) = 0. b) Use definition 6.1.1 to show that fis differentiable…

A: Solving by definition of 6.1.1 we have

Q: b) Let f(x) = x²-1 2-1 ' √x-1 0 < x < 1 1<x<3 2-1 ' 5x+3, 3x Find lim, 1 f(x). 3x-7 c) Let f(x) =…

A: The b part of the question can be solved using the concept of limits.The c part of the question can…

Q: Prove directly from definition that if f(x) and lim √(x) = ∞0 tim X-> 1 + then also lim f(x) = a ∞o…

A: A limit is said to be the value that a function approaches the output for the given input. The limit…

Q: (3) Evaluate lim -(1+sin(4))t dt

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Q: 2 A) lim f(x) = x²_1 X>-1 2 (x+1) ²

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Q: E< -2 Let F (x) os (z + 2) -2< a' Cos Find lim F(z). E→-2+ lim F(z) = z→-2+

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Q: If f(x) = 2x? – x + 2 , what is lim 2x2 – x + 2, lim 2x2 – x + 2, andf(2)? Is f(x)continuous at 2? O…

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Q: а) The function f is defined by 2, if x2 i) Find lim f(x). Give your reason.

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Q: Q.No 3.) if x<-1 Let f(x)={ax+ B if -13x<1 x² +2 if x21 Find a and B, so that lim f(x) and lim f…

A: Given that, Consider, limx→-1-f(x) = limx→-1- x3 = (-1)3 = -1 limx→-1+f(x) = limx→-1+ (αx+β)=…

Q: Find the value of lim f(x), if f(x): I6+ S8x +5 6 %3D O 52 О 16 O 20 О 32 O 10

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