Limits And Derivatives CBSE Questions & Answers

Limits And Derivatives

This is Mathematics Class 11 Limits and Derivatives CBSE Questions & Answers. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
\(\mathop {Lt}\limits_{x \to {\pi \over 4}} {{\sec x - \sqrt 2 } \over {x - {\pi \over 4}}}\) is equal to
  • A
    0
  • B
    \(\sqrt 2 \)
    Correct
  • C
    \(\sqrt 2 \)
  • D
    none of these
2
\(\mathop {Lt}\limits_{x \to {\pi \over 3}} \;\;{{\sec x - 2} \over {x - {\pi \over 3}}}\) is equal to
  • A
    \(2\sqrt 3 \)
    Correct
  • B
    \(\sqrt 3 \)
  • C
    2
  • D
    2+ \(\sqrt 3 \)
3
\(\mathop {Lt}\limits_{x \to \infty } \;\;\left( {\sqrt {{x^2} + x + 1} - x} \right)\) is equal to
  • A
    none of these
  • B
    2
  • C
    \({1 \over 2}\)
    Correct
  • D
    0
4
\(\mathop {Lt}\limits_{x \to \infty } \;\;\left( {\sqrt {{x^2} + 1} - x} \right)\) is equal to
  • A
    none of these
  • B
    2
  • C
    0
    Correct
  • D
    \({1 \over 2}\)
5
\(\mathop {Lt}\limits_{x \to {\pi \over 4}} \;{{\tan x - 1} \over {x - {\pi \over 4}}}\) is equal to
  • A
    0
  • B
    1
  • C
    2
    Correct
  • D
    \({1 \over 2}\)
6
\(\mathop {Lt}\limits_{h \to 0} \;{{\sin \sqrt {x + h} - \sin \sqrt x } \over h}\) is equal to
  • A
    \({1 \over {2\sin \sqrt x }}\)
  • B
    \(\sin \sqrt x \)
  • C
    \({{\cos \sqrt x } \over {2\sqrt x }}\)
    Correct
  • D
    \({{\cos \sqrt x } \over {2\sqrt x }}\)
7
\(\mathop {Lt}\limits_{x \to 1} \;\left( {\cos \left[ x \right]} \right)\)
  • A
    does not exist
    Correct
  • B
    is equal to 1
  • C
    none of these
  • D
    is equal to cos 1
8
\(\mathop {Lt}\limits_{x \to 0} \;{{\tan x} \over {\log (1 + x)}}\) is equal to
  • A
    does not exist
  • B
    none of these
  • C
    1
    Correct
  • D
    0
9
If \(f(x) = \int {{{x + \sin x} \over {x + \cos x}}dx},then \) \(\mathop {Lt}\limits_{x \to \infty } f'(x) = \)
  • A
    1
    Correct
  • B
    0
  • C
    \(\infty \)
  • D
    none of these
10
\(\mathop {Lt}\limits_{x \to 0} \;{\left( {{{1 + \tan x} \over {1 - \tan x}}} \right)^{{1 \over x}}}\)
  • A
    1
  • B
    0
  • C
    none of these
  • D
    \({e^2}\)
    Correct
11
\(\mathop {Lt}\limits_{x \to 0} \;{{\sin {x^n}} \over {{{\left( {\sin x} \right)}^m}}},n > m > 0,\) is equal to
  • A
    1
  • B
    0
    Correct
  • C
    \({m \over n}\)
  • D
    \({n \over m}\)
12
\(\mathop {Lt}\limits_{x \to 4} \;\;\;{{3 - \sqrt {5 + x} } \over {1 - \sqrt {5 - x} }} = \)
  • A
    does not exist
  • B
    \({1 \over 3}\)
  • C
    \( - {1 \over 3}\)
    Correct
  • D
    0
13
If G (x) = \(\sqrt {25 - {x^2}} \) then \(\mathop {Lt}\limits_{x \to 1} \;\;\;{{G(x) - G(1)} \over {x - 1}}\) has the value
  • A
    \({1 \over 24}\)
  • B
    \( - \sqrt {24} \)
  • C
    \({1 \over 5}\)
  • D
    \({1 \over {\sqrt {24} }}\)
    Correct
14
If \(y = \sqrt {\log x + \sqrt {\log x + \sqrt {\log x + ...to\infty } } } \) then \({{dy} \over {dx}}\) is equal to
  • A
    none of these
  • B
    \({x \over {2y - 1}}\)
  • C
    \({1 \over {x(2y - 1)}}\)
    Correct
  • D
    \({{\log x} \over {2y - 1)}}\)
15
If f be a function such that f (9) = 9 and f ‘ (9) = 3, then \(\mathop {Lt}\limits_{x \to 9} {{\sqrt {f(x)} - 3} \over {\sqrt x - 3}}\) is equal to
  • A
    none of these
  • B
    1
  • C
    3
    Correct
  • D
    9