Let the function f: R R be differentiable at x = 0. Prove that lim %3D = 0.

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: Use the limit definition to compute f '(a) and find an equation of the tangent line.

A: Answer: f'(-2)=-1

Q: compute f ' (a) using the limit definition and find an equation of the tangent line to the graph of…

A: Plug x=4 to find f'(4) Answer: f'(a)= -1/16

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&lt;x&lt;32 we have to prove that limx→16f(x)=4

Q: Show that the limit of the function f(x,y)= (0,0). (x+y)² x² + y² is indeterminate at

A: Limit of the function

Q: Sa F(t)dt 8. Suppose f is a differentiable function, f(a) = 0 and f'(a) # 0. Find lim f(x) if it…

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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…

Q: lim x* cos 1 = 0 .5

A: Show that , limx→0x4cos1x5=0

Q: x² – 6x, { 21? — 3и — 4, 1 4.

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Q: Show that the function f(x, y) x4 + y? has no limit as (x, y) approaches (0,0).

A: Note: Since we can solve at most one problem at a time, we have solved the first one. Kindly…

Q: Use the Squeeze Theorem to prove that lim (r.y)→(0,0) x +y² 0.

A: Using Squeeze Theorem to prove that lim(x,y)→(0,0)x4yx4+y2

Q: Find the linearization L(x) of f(x)=tan x at x=3pie/4.

A: Given

Q: Compute f'(a) in two ways, using Eq.(1) and Eq.(2). f(x) — 6х3, а %— 2 lim h→0 f(a + h) – f(a) h…

A: Given that f(x) = 6x3  , a =2we have to find f'(a) = limh→0 f(a+h)-f(a)hfirst find f(a+h)…

Q: Evaluate the limit lim (x,y)→(1,1) V(a – y). sin(x – y) x2 – y2

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Q: Let f(x) = V1– cos 2x a) f has a • discontinuity at 0. b) lim f(x) = c) lim f(x) =

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Q: Give a pair of functions f : R –→ R and g : R → R such that g is not tangent |f (x)-g(x)| || to f at…

A: To give a pair of function f(x) and g(x) such that g is not tangent to f at 0 and yet…

Q: use the limit definition to compute f'(a) and find an equation of the tangent line. f (x) = x3 + x,…

A: For a function f(x) the limit definition for derivative at x=a is f'(a)=limh→0f(a+h)-f(a)h. The…

Q: Find the limit and discuss the continuity of the function.

A: *The given function is,  f(x,y)=cos-1xy1+xy *The point at which the limit of the function is to be…

Q: Use the Squeeze Theorem to fĩnd lim In(x + 1) cos(-)

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Q: Evaluate the limit ´sin(t) -i + In(t + 5)j + (t + 5)* lim t+0 t Enter your answer in i, j, k form.

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Q: Let f(x) be a continuous differentiable function that f(x)- 4 satisfies lim =D5 %3D x3 x-3 Which…

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Q: lim, - f{x)= 0, then f(x)dx diverges Select one: O True False

A: When the limit exist or equal to 0 then the series is convergent otherwise divergent.

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

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Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f(x) = 2x 2 +…

A: Given:                fx=2x2+10x,     a=3 Find: f3+h and f3 f3+h=23+h2+103+h…

Q: Complete the table and use the result to estimate the limit. cos(-5x)-1 lim -5x -0.1 -0.01 -0.001…

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Q: Use the graphs of f and g and their tangent lines at (-1,0) to find f(x) lim Number x-1 g(x) y f(x)…

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Q: Use the graphs of f and g and their tangent lines at (2, 0) to find lim f(x) x+2 g(x) y4 y = 1.6(x -…

A: We need to evaluate the value of limx→2f(x)g(x) using the following graph

Q: Find the limits in Exercises

A: Given: limx→∞x57+x-3x57+cos x   ……1

Q: Evaluate , x? dx by limit definition. f(x) dx f(G.) Ax¡ lim i=1

A: Given integral  ∫01x2dx △x=b-an△x=1-0n=1n ci=a+i(△x)ci=0+i1nci= inf(ci)=in2

Q: Use the limit definition to compute f 1 (a) and find an equation of the tangent line. f (x) = x +…

A: For a function f(x) the limit definition for derivative at x=a is f'(a)=limh→0f(a+h)-f(a)h. For a…

Q: Let y(t) be a solution of ý = y(1 - ;) such that y(0) = 4. Determine lim y(t) without finding y(t)…

A: Given that y(t) is the solution of y.=12y(1-y2)with y(0)=4To find the limitlimt→∞y(t)

Q: Determine the set of points at which the function F(x, y) = arctan ( 3x +7/y – 7 ) is continuous.

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Q: Consider the function f (x, y) : 2y Show that lim (x,y)→(0,0) f (x, y) does not exist by ɛ – 8…

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Q: Find the limits in Exercises 19–22. dx tan-t dt 19. lim b→1", 20. lim VI - x² 21. lim 2n n + 2 n-00…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: x-y (x,y) →(0.0) x + y 2.) Prove that lim does not exist.

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Q: (b) Let h(r, y) be a function defined by h(r, y) = h(0, 0) = 0 3r2 + y? (i) Prove that (ii) Using…

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Q: Find the limit. lim (х, у) > (1, 55) Vx + у Discuss the continuity of the function. continuous for x…

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Q: d. lim x²(ln x)² x→0+ e. lim (cos x)'1/x X→0

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Q: Use the Squeeze Theorem to prove that lim væ[1+ (sin(27/x))²] = 0.

A: "Since you have asked multiple questions in a single request, we will be answering only the 1st…

Q: Please

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Q: 1-cost lim sint et-1 + t t t+0

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Q: Use the limit definition to compute f '(a) and find an equation of the tangent line. f (x) = x−1, a…

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

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Q: Determine lim In/xy. (x,y)(e4)

A: lim(x,y)-&gt;(e2,4)lnxy

Q: Evaluate the limid lim X'sin t X→o sin x?

A: We have to find limit

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

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Q: Let sin(2t) F(t) = + cos t, Find the limit lim F(t).

A: We have to find the limit

Q: d. Use the limit definition of the derivative, f (x) = lim f (x+h)-f(x) 1 +h)=j, to prove that VT =…

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Q: If f' is continuous on [0, ∞) and limx→∞ f (x) = 0 , showthat

A: Given, If f' is continuous on the interval [0,∞) and limx→∞f(x)=0 show that, ∫0∞f'(x)dx=-f(0)