Q1. Find the value of

x + 8 sinx/ 4x + 10

Solution.
As we know that, -1 ≤ sin x ≤ 1 for all x

-8 ≤ 8 sinx ≤ 8

x - 8 ≤ x + 8 sinx ≤ x + 8

dividing all terms by 4x + 10

x - 8/ 4x + 10

≤

x + 8 sinx/ 4x + 10

≤

x + 8/ 4x + 10

x + 8/ 4x + 10

=

1 - 8/x/ 4 + 10/x

=

1/ 4

So,

x + 8 sinx/ 4x + 10

=

1/ 4

Q2. Find the value of x^1/1-x

indeterminate form of 1

log y =

log x/ 1 - x

log y = -1 as x → ∞

y = 1/e

∴

log x/ 1 - x

= -1

Q3. Find the value of

4x⁴ - 3x + 6/ x⁴ - 6x² + 2

Solution.

4 - 3/x³ + 6/x⁴/ 1 - 6/x² + 2/x⁴

= 4

Q4. Find the value of

(x - 4)/ lx - 4l

(a) 1 (b) 2 (c) 0 (d) limit does not exist

Solution.

(x - 4)/ -(x - 4)

= -1

(x - 4)/ (x - 4)

= 1

LHL ≠ RHL , So limit does not exist

Q5. Find the value of

(1 + x)⁵ - 1/ x

Solution.
Put 1 + x = y

x = y - 1

x → 0 , y → 1

=

y⁵ - 1/y - 1

= 5(1)^{5 - 1} = 5

Q6. Find the value of

e^x - e⁴/ x - 4

Solution.
Apply D'Hospital law, So derivate the terms

e^x - 0 /1 - 0

= e^x = e⁴

Q7. Evaluate

e^-x - e^-1/ x - 1

Solution.
Apply D'Hospital law,

- e^-x = -

1/e³

Q8. Evaluate

√(5x⁴ - 6x² + 9x + 10)/ 4x²

Solution.

√(5 - 6/x² + 9/x³ + 10/x⁴) / 4

=

√5/ 4

Q9. Evaluate

4x + lxl/ 9x - 7lxl

(a) 1 (b) 1/2 (c) 5/2 (d) limit does't exist

Solution.
LHL =

4x - x/ 9x + 7x

=

3x/ 16x

=

3/ 16

RHL =

4x + x/ 9x - 7x

=

5x/ 2x

=

5/ 2

LHL ≠ RHL

Q10. Evaluate

(n + 2)! + (n + 1)!/ (n + 2)! - (n + 1)!

Solution.
=

(n + 2)(n + 1)! + (n + 1)!/ (n + 2)(n + 1)! - (n + 1)!

=

(n + 1)! (n + 2 + 1)/ (n + 1)! (n + 2 - 1)!

=

n + 3/ n + 1

=

1 + 3/4/ 1 + 1/4

= 1