Use an appropriate definition to prove the following limits. x³ – 16x* = 1 I+0 r°+ 4x² x +2 V9x2 + 4 (a) lim z→5+ x² – 25 (b) lim (c) lim -3 %3D x +2

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A: This question is from real analysis & our topic is uniform continuity .

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A: topic - derivative of a function using limits

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A: x=yx=1000y2Solve both the equationsy=1000y2y3=1000y=10The limits of y are from 10 to 20.

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A: Given that     D(x) = -3x/10 + 8     S(x) = x/5 + 3

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A: Hee, the given surface is z=7-x2-y2 and the plane x+3y+8z=0 which passes through the origin. So, the…

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A: Using substitution ,

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A: Using given formula,

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A: To find linear approximation of given function

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Q: 4. Find the exact value of the expression. (a) 10log 36 (b) In e3

A: Explanation of the answer is as follows

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Q: 2. Solve x? – 6ax+ 2 = 0, where a>1 is a constant. -

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Q: 2t2 + 3, y = 6t – 1 at the point whose 3. Find the equation of the normal to the curve defined by x…

A: Explanation of the answer is as follows

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A: We can solve this as follows:

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A: To find the area of the region enclosed between the two curves.

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A: Given: xlny=zArctan3zy

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A: LetI=∫lnz3zdzLet z3=udifferentiate 3z2dz=duDivide 3z2 on both sidedz=du3z2dz=du3z2Divide z oboth…

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A: Solved below

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Q: 9. Find the sum. (a) 22 32 23 33 29 34 310 (b) 1 + 2/2 23/2 11 1/2 1.

A: b)1+1212+12+1232+.........

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A: Given:  An ellipse:x252+y242=1  , rotated around x-axis. A cylinderical hole of radius 2cm along…

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Q: Q10. For the function f(x, y) = x³e³y_find the maximum value of its directional derivative at the…

A: Here, f(x,y)=x3e3y⇒∇f=∂f∂x,∂f∂y=3x2e3y,3x3e3y⇒∇f(1,0)=3(1)e0,3(1)e0=(3,3)