2) If 2 – x2 < f (x) < 2+ 3x² whenever -1 < x < 1, use limit theorems to find lim0 f (x). Give all your reasons.

Q: If x3 f(x) s x for x in [-1,1], find lim f(x) if it exists. X-0

A:

Q: Formally prove that lim 2x+3=7 using the ɛ,8 definition of a limit. x→2

A:

Q: lim (4.x – 3) = 9

A: The given limit,      limx→34x-3=9

Q: Use l’Hospital’s rule where applicable to find the limit.

A: the given limit is: limx→08x6+3x4-9x9x7-2x4+x3 we have to evaluate the given limit.

Q: Calculate the limit lim 2 / 5x-15 as x goes to 3^+

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Q: Find the limit, if it exists. 29) lim x+12x2 -5x 5x To X-0

A: First we have to factor out x from the numerator.

Q: Find the limit, if it exists.

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Q: Use the precise definition of a limit to prove lim Vx – 5 = 2 x-9

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Q: x² – 1 lim x→1 x – 1

A:

Q: Find the limits x² - 7x lim X-o X + 1

A: Given: limx→∞x2-7xx+1

Q: Find the indicated if it exists: lim x to 8;  2x2+3x+1 divided by x4 - x2

A: To find the limit if exists

Q: ex ) 4. Use the definition of the limit to prove that lim 4x + 2 = 14. %3D x-3

A:

Q: Compute the indicated limit. 5x2 – 5x – 3 lim x00 4x – 2x? – 7

A:

Q: (b) Prove that lim,→20,¤€A ƒ (x) exists and is equal to lim→0 Yk-

A: This problem is related to the sequence of functions. Given:  fkk∈ℕ is a sequence of functions from…

Q: use definition of limit to show:   lim    (4x + 1) =13 x-&gt;3

A: The definition of limit:

Q: Find the limit.   Lim  x→1+      x1/1-x

A: Given function is By property of logarithm

Q: Find Limit Using Squeeze Theorem: 2. If 2x – 2s f(x)<x² – 4x + 3 for x 2 0, find lim f(x).

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Q: Evaluate the limit. lim x² In x x→0+

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Q: lim(4x – 3) = 9 x→3

A:

Q: Find the value of the limit: 3 tan(4x)– 12x lim

A: From the given problem:   limx→03tan4x-12xx3   This problem can be solved by the series expansion of…

Q: Find the limit. lim x2/(1-x) X→1 *

A:

Q: 2. If it exists, find lim f(x) where X--1 x =-1 f(x) = { *x– 1)³, x+-1 4, If the limit does not…

A:

Q: 10. Use the definition of limit to show that a. lim (x² + 4x) = 12, Z-2

A:

Q: If Lim [x3 – g(x)] = 12 as x tends to 3, use the Rules of Limits to find Lim g(x) as x tends to 3.…

A: Given: limx→3x3-gx=12

Q: Prove the statement using the ɛ, d definition of a limit. 1. lim (x² – 4x +5) = 1 x→2 2. lim (x² –…

A: use the definition of epsilon and delta definition  to prove the limit.

Q: 2x-1 2. lim X-2 \3x+4,

A:

Q: 4. Using the precise definition of a limit, prove lim (2x +8) = 38. x-15

A:

Q: Evaluate the limit, if it exists.

A: We have to evaluate the limit: limx→164-x16x-x2 It is 00 form, since there is square root function…

Q: verify that L’Hôpital’s Rule applies and evaluate the limit.

A:

Q: find the one-sided limit ( if it exists)

A: The given limit is,

Q: Find the limit. x2 - 8x + 15 x3 - 9x lim - 00 O 15 Does not exist 8

A: First factor the denominator and use limit of two sided limit approach,

Q: Estimate the limit numerically for lim X4 x²-3x-4 4 3.9 3.99 3.999 4.001 4.01 4.1 X. x - 4 Lx2 - 3x…

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Q: Prove that lim (2x – 7) = 3 by using the e, 8 definition of a limit. x→5

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Q: Evaluate the limit. 5 sin(x) – 5x lim x3

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Q: Use L'Hôpital's rule to find the limit. 3x - 1 lim 3 x³ + 4

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Q: Use L'Hospital's Rule (if appropriate) to find the limit: x² - 7x +12 lim- x² - 9 X→3

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Q: Evaluate the limit  Limx-&gt;0 (ex-1-x)/x3

A:

Q: Which property of f(x) = x' allows us to conclude that lim x- 8?

A: f(x)=x3 is a continuous function. So, it is continuous at x=2 also.

Q: lim(4x – 3) = 5 x→2

A:

Q: Evaluate the limit 5 + 2x lim 8 – 8x x → 0

A: We haven the function is: f(x) = (5 + 2x)/(8 - 8x)  Dividing the numerator and the denominator by x…

Q: Compute the indicated limit. „2 x + 3 lim x00 x - 4

A:

Q: Find the limit. x* – 1 - lim x→0+ In (x) +x – 1 п lim ( 1+ n

A: These question is based on Limits.

Q: —ах+6 if x2-1 Let f(x)=< For what value of a does 2x2 +ах ifx<-1 lim f(x) exist? x--1

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Q: 5, evaluate the limit. 2x3-x 5) lim X0 x2+3x-1

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Q: If + 1for all positive x-values, then what is lim f(x)? X- 4

A: We have to find limit

Q: Use the formal definition of the limit to prove.  lim (4x − 3) = −7 x→−1

A:

Q: Let f(x) = x^2 − 10x. If you construct an ϵ-δ proof to show that limx→5 f(x) = −25, what is the…

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Q: Evaluate the limit. x2 + 2x – 15 lim- x – 3 X-3

A:

Family of Curves

A family of curves is a group of curves that are each described by a parametrization in which one or more variables are parameters. In general, the parameters have more complexity on the assembly of the curve than an ordinary linear transformation. These families appear commonly in the solution of differential equations. When a constant of integration is added, it is normally modified algebraically until it no longer replicates a plain linear transformation. The order of a differential equation depends on how many uncertain variables appear in the corresponding curve. The order of the differential equation acquired is two if two unknown variables exist in an equation belonging to this family.

XZ Plane

In order to understand XZ plane, it's helpful to understand two-dimensional and three-dimensional spaces. To plot a point on a plane, two numbers are needed, and these two numbers in the plane can be represented as an ordered pair (a,b) where a and b are real numbers and a is the horizontal coordinate and b is the vertical coordinate. This type of plane is called two-dimensional and it contains two perpendicular axes, the horizontal axis, and the vertical axis.  

Euclidean Geometry

Geometry is the branch of mathematics that deals with flat surfaces like lines, angles, points, two-dimensional figures, etc. In Euclidean geometry, one studies the geometrical shapes that rely on different theorems and axioms. This (pure mathematics) geometry was introduced by the Greek mathematician Euclid, and that is why it is called Euclidean geometry. Euclid explained this in his book named 'elements'. Euclid's method in Euclidean geometry involves handling a small group of innately captivate axioms and incorporating many of these other propositions. The elements written by Euclid are the fundamentals for the study of geometry from a modern mathematical perspective. Elements comprise Euclidean theories, postulates, axioms, construction, and mathematical proofs of propositions.

Lines and Angles

In a two-dimensional plane, a line is simply a figure that joins two points. Usually, lines are used for presenting objects that are straight in shape and have minimal depth or width.