JEE Mathematics Limits Indeterminate Forms MCQs Set 02

Choose the most appropriate option (a, b, c or d).

Question. If f(4) = f, f'(4) = 1 then \(\lim _{x \rightarrow 4} \frac{2-\sqrt{f(x)}}{2-\sqrt{x}}\) is equal to
(a) 0
(b) 1
(c) -1
(d) None of the options
Answer: (b) 1

Question. The graph of the function y = f(x) has a unique tangent at the point (a, 0) through which the graph passes. Then \(\lim _{x \rightarrow a} \frac{\log _{e}\{1+6 f(x)\}}{3 f(x)}\) is
(a) 1
(b) 0
(c) 2
(d) None of the options
Answer: (c) 2

Question. Let f(x) be a twice-differentiable function and f''(0) = 2 then \(\lim _{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4 x)}{x^{2}}\) is
(a) 6
(b) 3
(c) 12
(d) None of the options
Answer: (a) 6

Question. If f(x), g(x) be differentiable functions and f(1) = g(1) = 2 then \(\lim _{x \rightarrow 1} \frac{f(1) g(x)-f(x) g(1)-f(1)+g(1)}{g(x)-f(x)}\) is equal to
(a) 0
(b) 1
(c) 2
(d) None of the options
Answer: (c) 2

Question. If f(a) = 2, f'(a) = 1, g(a) = -1, g'(a) = 2 then \(\lim _{x \rightarrow a} \frac{g(x) f(a)-g(a) f(x)}{x-a}\) is
(a) -5
(b) \(\frac{1}{5}\)
(c) 5
(d) None of the options
Answer: (c) 5

Question. \(\lim _{n \rightarrow \infty}\left(1+\sin \frac{a}{n}\right)^{n}\) is equal to
(a) \(e^{a/2}\)
(b) \(e^{a}\)
(c) \(e\)
(d) \(e^{2a}\)
Answer: (b) \(e^{a}\)

Question. \(\lim _{x \rightarrow 0}\left(\frac{1+5 x^{2}}{1+3 x^{2}}\right)^{1 / x^{2}}\)
(a) \(e\)
(b) \(e^{1/2}\)
(c) \(e^{-2}\)
(d) None of the options
Answer: (d) None of the options

Question. \(\lim _{x \rightarrow 0}\left\{\tan \left(\frac{\pi}{4}-x\right)\right\}^{1 / x}\) is equal to
(a) 1
(b) \(e\)
(c) \(e^{2}\)
(d) \(e^{-2}\)
Answer: (d) \(e^{-2}\)

Question. \(\lim _{x \rightarrow \pi / 4}(2-\tan x)^{\log \tan x}\) is equal to
(a) 0
(b) 1
(c) \(e\)
(d) \(e^{-1}\)
Answer: (b) 1

Question. \(\lim _{x \rightarrow \infty}\left(\frac{x-1}{x+1}\right)^{x+2}\) is equal to
(a) \(e\)
(b) \(e^{-1}\)
(c) \(e^{-2}\)
(d) None of the options
Answer: (c) \(e^{-2}\)

Question. If \(\lim _{x \rightarrow \infty}\left(1+\frac{\lambda}{x}+\frac{\mu}{x^{2}}\right)^{2 x}=e^{2}\) then
(a) \(\lambda = 1, \mu = 2\)
(b) \(\lambda = 2, \mu = 1\)
(c) \(\lambda = 1, \mu = \text{any real constant}\)
(d) \(\lambda = \mu = 1\)
Answer: (c) \(\lambda = 1, \mu = \text{any real constant}\)

Question. \(\lim _{\theta \rightarrow 0^{+}} \frac{\sin \sqrt{\theta}}{\sqrt{\sin \theta}}\) is equal to
(a) 0
(b) 1
(c) -1
(d) None of the options
Answer: (b) 1

Question. Let {x} denote the fractional part of x. Then \(\lim _{x \rightarrow 0} \frac{\{x\}}{\tan \{x\}}\) is equal to
(a) 1
(b) 0
(c) -1
(d) None of the options
Answer: (a) 1

Question. \(\lim _{x \rightarrow \infty} \frac{\log _{e}[x]}{x}\), where [.] denotes the greatest integer function, is
(a) 0
(b) 1
(c) -1
(d) nonexistent
Answer: (a) 0

Question. \(\lim _{x \rightarrow 1} \frac{\sqrt{1-\cos 2(x-1)}}{x-1}\)
(a) exists and it is \(\sqrt{2}\)
(b) exists and it is \(-\sqrt{2}\)
(c) does not exist because \(x-1 \rightarrow 0\)
(d) does not exist because LH lim \(\neq\) RH lim
Answer: (d) does not exist because LH lim \(\neq\) RH lim

Question. \(\lim _{x \rightarrow 1} \frac{x \sin \{x-[x]\}}{x-1}\), where [.] denotes the greatest integer function, is
(a) 0
(b) -1
(c) not existent
(d) None of the options
Answer: (c) not existent

Question. \(\lim _{x \rightarrow 0} \frac{x[x]}{\sin |x|}\), where [.] denotes the greatest integer function, is
(a) 0
(b) 1
(c) not existent
(d) None of the options
Answer: (c) not existent

Question. \(\lim _{x \rightarrow 2}\{[2-x]+[x-2]-x\}\)
(a) is 0
(b) is 3
(c) is -3
(d) does not exist
Answer: (c) is -3

Question. \(\lim _{x \rightarrow-1}\{[x]+|x|\}\), where [.] denotes the greatest integer function,
(a) is 0
(b) is 1
(c) does not exist
(d) None of the options
Answer: (c) does not exist

Question. If \(f(x) = \frac{\sin[x]}{[x]}, [x] \neq 0\); \(0, [x] = 0\), where [.] denotes the greatest integer function, then \(\lim _{x \rightarrow 0} f(x)\) is equal to
(a) 1
(b) 0
(c) -1
(d) None of the options
Answer: (d) None of the options

Question. Let f(x) = \(x^{2}-1, 0
(a) \(x^{2}-6 x+9=0\)
(b) \(x^{2}-10 x+21=0\)
(c) \(x^{2}-14 x+49=0\)
(d) None of the options
Answer: (b) \(x^{2}-10 x+21=0\)

Question. If [.] denotes the greatest integer function then \(\lim _{n \rightarrow \infty} \frac{[x]+[2 x]+\dots+[n x]}{n^{2}}\) is
(a) 0
(b) x
(c) \(\frac{x}{2}\)
(d) \(\frac{x^{2}}{2}\)
Answer: (c) \(\frac{x}{2}\)

Question. \(\lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \cos t^{2} d t}{x \sin x}\) is equal to
(a) 1
(b) 2
(c) 0
(d) None of the options
Answer: (a) 1

Question. \(\lim _{x \rightarrow a} \frac{x}{x-a} \cdot \int_{a}^{x} f(x) d x\) is equal to
(a) f(a)
(b) af(a)
(c) 0
(d) None of the options
Answer: (b) af(a)

Question. \(\lim _{x \rightarrow 1+0} \frac{\int_{1}^{x}|t-1| d t}{\sin (x-1)}\) is equal to
(a) 0
(b) 1
(c) -1
(d) None of the options
Answer: (a) 0

Question. Let f(x) = sin x, \(x \neq n \pi\); 2, \(x = 2 \pi\), where \(n \in \mathbb{Z}\). Then \(\lim _{x \rightarrow 0} g(f(x))\) is
(a) 0
(b) 1
(c) 3
(d) None of the options
Answer: (b) 1

Question. If f(x) continuous in [0, 1] and \(f\left(\frac{1}{3}\right) = 1\) then \(\lim _{n \rightarrow \infty} f\left(\frac{n}{\sqrt{9 n^{2}+1}}\right)\) is
(a) 1
(b) 0
(c) \(\frac{1}{3}\)
(d) None of the options
Answer: (a) 1

Question. If f(x) is continuous and \(f\left(\frac{9}{2}\right) = \frac{2}{9}\) then \(\lim _{x \rightarrow 0} f\left(\frac{1-\cos 3 x}{x^{2}}\right)\) is equal to
(a) \(\frac{9}{2}\)
(b) \(\frac{2}{9}\)
(c) 0
(d) None of the options
Answer: (b) \(\frac{2}{9}\)

Choose the correct options. One or more options may be correct.

Question. Let f(x) = \(1+\frac{2 x}{a}, 0 \leq x<1\); ax, \(1 \leq x<2\). If \(\lim _{x \rightarrow 1} f(x)\) exists then a is
(a) 1
(b) -1
(c) 2
(d) -2
Answer: (b) -1, (c) 2

Question. If \(\alpha\) is a repeated root of \(a x^{2}+b x+c=0\) then \(\lim _{x \rightarrow \alpha} \frac{\sin \left(a x^{2}+b x+c\right)}{(x-\alpha)^{2}}\) is
(a) 0
(b) a
(c) b
(d) c
Answer: (b) a

Question. If \(f(x)=|x-1|-[x]=\) the greatest integer less than or equal to x, then
(a) f(1 + 0) = -1, f(1 - 0) = 0
(b) f(1 + 0) = 0 = - f(1 - 0)
(c) \(\lim _{x \rightarrow 1} f(x)\) exists
(d) \(\lim _{x \rightarrow 1} f(x)\) does not exist
Answer: (a) f(1 + 0) = -1, f(1 - 0) = 0, (d) \(\lim _{x \rightarrow 1} f(x)\) does not exist

Question. If \(\lim _{n \rightarrow \infty}\left(a n-\frac{1+n^{2}}{1+n}\right)=b\), a finite number, then
(a) a = 1
(b) a = 0
(c) b = 1
(d) b = -1
Answer: (a) a = 1, (c) b = 1

Question. Let \(\tan \alpha \cdot x+\sin \alpha \cdot y=\alpha\) and \(\alpha \operatorname{cosec} \alpha \cdot x+\cos \alpha \cdot y=1\) be two variable straight lines, \(\alpha\) being the parameter. Let P be the point of intersection of the lines. In the limiting position when \(\alpha \rightarrow 0\), the point P lies on the line
(a) x = 2
(b) x = -1
(c) y + 1 = 0
(d) y = 2
Answer: (a) x = 2, (c) y + 1 = 0