This set of Engineering Mathematics online test focuses on “Limits and Derivatives of Several Variables – 3”.

1. lim_{x → 1} (x-1)Tan(^{πx}⁄_{2}) is

a) 0

b) –^{1}⁄_{π}

c) –^{2}⁄_{π}

d) ^{2}⁄_{π}

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2. Value of limit always be in the range of function?

a) True

b) False

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Explanation: Because the range of f(x) = {x} is [0,1) and it value at lim

_{x → 1} – f(x) is 1 which is not in its range.

3. Necessary Conditions of Sandwich rule is

a) All function must have common domain.

b) All function must have common range.

c) All function must have common domain and range both..

d) Function must not have common domain and range.

View Answer

Explanation: Statement of sandwich theorem is, If Functions f(x) ,g(x) and h(x)

1. have Common Domain,

2. and, satisfy f(x) ≤ g(x) ≤ h(x) ∀ x ∈ D

Then if f(x) = h(x) = L

=> g(x) = L .

4. The value of lim_{x → 0} [x]Cos(x), [x] denotes the greatest integer function

a) lies between 0 and 1

b) lies between -1 and 0

c) lies between 0 and 2

d) lies between -2 and 0

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Explanation: lim

_{x → 0} [x]Cos(x)

We know that,

x-1 < [x] < x

Multiplying by Cos(x), we get

(x-1)Cos(x) < [x]Cos(x) < xCos(x)

Taking limits, we get

lim_{x → 0} [(x-1)Cos(x)] < lim_{x → 0} [x]Cos(x) < lim_{x → 0}[xCos(x)]
=> -1 < lim_{x → 0} [x]Cos(x) < 0.

5. Value of lim_{x → 0}[(1+xe^{x} )/(1 – Cos(x))]
a) e

b) 1

c) 2

d) Can not be solved

View Answer

Explanation: =>lim

_{x → 0}[(1+xe

^{x})/(1 – Cos(x))] =

^{1}⁄

_{0}(Indeterminate)

=> By L’Hospital rule

=> lim

_{x → 0}[(1+xe

^{x}) / (Sin(x))] =

^{1}⁄

_{0}(Again indeterminate)

=> By L’ Hospital rule

=> lim

_{x → 0}[((2+x)e

^{x})/ (Cos(x))] = 2.

6. The value of , [x] denotes the greatest integer function

a) 0

b) 1

c) ∞

d) – ∞

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7. Evaluate lim_{x → 0}(1+Tan(x))^{Cot(x)}

a) 1

b) e

c) ln(2)

d) e^{2}

View Answer

Explanation:

lim

_{x → 0}(1+Tan(x))

^{Cot(x)}= lim

_{tan(x) → 0}(1+Tan(x))

^{1⁄Tan(x)}= lim

_{t → 0}(1 + t)

^{1⁄t}= e.

8. Evaluate lim_{x → 1}[(-x^{x} + 1) / (xlog(x)) ]
a) e^{e}

b) e

c) -1

d) e^{2}

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9. Find domain of n for which lim_{x → 0}e^{nx}Cot(nx) , has non zero value.

a) n ∈ (0,∞) ∩ (1,5)

b) n ∈ (-∞,∞) ∩ (1,5)

c) n ∈ (-∞,∞)

d) n ∈ (-∞,∞) ~ 5

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10. Value of (dSin(x)Cos(x)) / dx is

a) Cos(2x)

b) Sin(2x)

c) Cos^{2}(2x)

d) Sin^{2}(2x)

View Answer

Explanation: (dSin(x)Cos(x)) / dx=Cos(x) dSin(x)/dx + Sin(x) dCos(x)/dx = Cos

^{2}(x) – Sin

^{2}(x) = Cos(2x).

11. Evaluate

a) 1

b) e

c) 0

d) e^{2}

View Answer

12. If , then find the value of a and b.

a) 2.5, -1.5

b) -2.5, -1.5

c) -2.5, 1.5

d) 2.5, 1.5

View Answer

Explanation

Since, given limit is finite, hence coefficients of powers of x should be zero and x

^{3}should be 1

⇒ 1 + a – b=0

⇒

^{b}⁄

_{6}–

^{a}⁄

_{2}= 1

⇒ Solving the above two equations we get, a = -2.5, b = -1.5.

13. , then find the value of a, b and c.

a) 1.37, -4.13, 4.13

b) 1.37, 4.13, -4.13

c) -1.37, 4.13, 4.13

d) 1.37, 4.13, 4.13

View Answer

Explanation:

Now, coefficient of x and x^3 should be zero and that of x

^{5}should be 1, then

⇒ B + c = 0

⇒

^{b}⁄

_{6}+

^{c}⁄

_{2}= a

⇒

^{b}⁄

_{120}+

^{c}⁄

_{24}= 1

⇒ By solving these 3 equations, a = 1.37, b = 4.13, c = -4.13.

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