JEE Mathematics Limits Continuity and Differentiability MCQs Set 09

Answer: (c) \(e^4\)

Explanation:
1. This is of the form \(1^\infty\): exponent \(\lim x\cdot\dfrac{4x+1}{x^2+x+2}=4\).
2. Limit \(=e^4\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (d) \(3\)

Explanation:
1. Both bounds tend to 3 as \(x\to0\).
2. By the sandwich theorem the limit is 3.

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) \(3\)

Explanation:
1. \(\cos x-1\sim-\dfrac{x^2}2\) and \(\cos x-e^x\sim-x\).
2. The numerator behaves like \(\dfrac{x^3}2\), so \(n=3\) (limit \(\dfrac12\)).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (b) \(1\)

Explanation:
1. \((\sin x)^{1/x}\approx x^{1/x}\to0\).
2. \(\left(\dfrac1x\right)^{\sin x}=e^{-\sin x\ln x}\to e^0=1\).
3. Sum \(=1\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (d) \(e^{-2/\pi}\)

Explanation:
1. Form \(1^\infty\): exponent \(\lim\tan\dfrac{\pi x}{2a}\left(1-\dfrac ax\right)\). Put \(x=a+h\).
2. \(\tan\left(\dfrac\pi2+\dfrac{\pi h}{2a}\right)=-\cot\dfrac{\pi h}{2a}\approx-\dfrac{2a}{\pi h}\) and \(1-\dfrac ax\approx\dfrac ha\).
3. Exponent \(=-\dfrac2\pi\); limit \(e^{-2/\pi}\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (b) \(n\)

Explanation:
1. As \(t=\dfrac1{\cos^2x}\to\infty\), the sum is dominated by \(n^t\).
2. \((n^t)^{1/t}\cdot\left(1+\dots\right)^{1/t}\to n\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) \(k\)

Explanation:
1. Using \(\log_ab=\dfrac{\ln b}{\ln a}\) the product telescopes to \(\dfrac{\ln n^k}{\ln(n-1)}\).
2. \(=k\dfrac{\ln n}{\ln(n-1)}\to k\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (a) \(e^{-a^2/b^2}\)

Explanation:
1. Form \(1^\infty\): exponent \(\lim\sec^2\left(\dfrac\pi{2-bx}\right)\left(\sin^2\dfrac\pi{2-ax}-1\right)=-\lim\dfrac{\cos^2\frac\pi{2-ax}}{\cos^2\frac\pi{2-bx}}\).
2. \(\cos\dfrac\pi{2-ax}\approx-\dfrac{\pi ax}4\) and \(\cos\dfrac\pi{2-bx}\approx-\dfrac{\pi bx}4\), so the ratio of squares tends to \(\dfrac{a^2}{b^2}\).
3. Limit \(e^{-a^2/b^2}\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (d) \(\dfrac ab\)

Explanation:
1. \(f(1-\cos x)\approx a(1-\cos x)^2\approx\dfrac{ax^4}4\), so \(g(x)\approx\dfrac{ax^4}{4bx^2}=\dfrac{ax^2}{4b}\).
2. \(g(1-\cos2x)\approx g(2x^2)\approx\dfrac{a(2x^2)^2}{4b}=\dfrac{ax^4}b\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) \(G.P.\)

Explanation:
1. The relation gives \(f(x)=1\pm x^n\); \(f(4)=65\) gives \(f(x)=x^3+1\).
2. \(f'(x)=3x^2\): if \(l_2^2=l_1l_3\) then \((3l_2^2)^2=3l_1^2\cdot3l_3^2\), so G.P.

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (b) 1

Explanation:
1. Form \(1^\infty\): exponent \(\lim\tan x(\sin x-1)=\lim\dfrac{\sin x(\sin x-1)}{\cos x}=0\).
2. So the limit is \(e^0=1\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (b) \(\dfrac12\)

Explanation:
1. Rationalise: \(\dfrac{\sqrt{x+\sqrt x}}{\sqrt{x+\sqrt{x+\sqrt x}}+\sqrt x}\).
2. Divide by \(\sqrt x\): numerator \(\to1\), denominator \(\to2\); the limit is \(\dfrac12\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) 6

Explanation:
1. \(f(x)\sim x\), so the numerator must be \(x^3+\dots\): \(x(1+a)-bx+\left(-\dfrac a2+\dfrac b6\right)x^3\).
2. \(1+a-b=0\) and \(-\dfrac a2+\dfrac b6=1\): \(a=-\dfrac52,\ b=-\dfrac32\).
3. \(b-3a=-\dfrac32+\dfrac{15}2=6\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (a) \(0\text{ if }n>m\)

Explanation:
1. \(\dfrac{\sin x^n}{(\sin x)^m}\sim x^{n-m}\).
2. It tends to 0 for \(n>m\), to 1 for \(n=m\) and is unbounded for \(n<m\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) 0

Explanation:
1. \(S_{n+1}-S_n=a_{n+1}\to a\).
2. \(\sqrt{\sum k}=\sqrt{\dfrac{n(n+1)}2}\to\infty\), so the limit is 0.

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (d) \(\dfrac{4^{n+1}-1}{3\cdot2^n}\)

Explanation:
1. \(a=2\) (at \(x=-1\)) and \(b=\dfrac12\).
2. \(\sum_{r=0}^n2^r2^{-(n-r)}=2^{-n}\sum4^r=\dfrac{4^{n+1}-1}{3\cdot2^n}\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (a) 1, 2 and 3

Explanation:
1. For \(x>0\): \(x^2-3x+2=(x-1)(x-2)\) is positive on \((0,1)\cup(2,\infty)\) and negative on \((1,2)\), so \(f'=2x-3\) on the first set: statement 1 holds.
2. For \(x<0\): \(x^2+3x+2=(x+1)(x+2)\) is positive on \((-\infty,-2)\cup(-1,0)\) (so \(f'=2x+3\)) and negative on \((-2,-1)\) (so \(f'=-2x-3\)).
3. All three statements are correct.

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (b) \(\dfrac32\)

Explanation:
1. For \(|x|<1\): \(f=x\); for \(|x|>1\): \(f=0\); \(f(1)=\dfrac12\).
2. \(f(1^+)=0,\ f(1^-)=1,\ f(1)=\dfrac12\); the sum is \(\dfrac32\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (d) \(f(x)=1\text{ for }|x|>1\text{ and }f(x)=-1\text{ for }|x|<1\)

Explanation:
1. For \(|x|>1\): \(x^{2n}\to\infty\) and \(f=1\).
2. For \(|x|<1\): \(x^{2n}\to0\) and \(f=-1\); at \(|x|=1\), \(f=0\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (a) 2

Explanation:
1. \(f'(x)=nx^{n-1}+1\), so the condition is \(n(x+y)^{n-1}=nx^{n-1}+ny^{n-1}\).
2. \(n=0\) works trivially (\(0=0\)); \(n=2\) works (\(x+y=x+y\)); \(n=1\) fails (\(1=2\)).
3. Sum \(=0+2=2\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) 1

Explanation:
1. \(=1-\dfrac{n\ln x}{[x]}\).
2. \(\dfrac{\ln x}{[x]}\to0\), so the limit is 1.

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (a) 1

Explanation:
1. \(y=\dfrac{1-x^{2^{n+1}}}{1-x}=1+x+x^2+\dots+x^{2^{n+1}-1}\).
2. \(y'(0)=1\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (b) \(-\dfrac13\)

Explanation:
1. Expand and set the coefficients of \(1,x,x^2\) to zero: \(a+b=0,\ 1+a-b+c=0,\ a+b-c=0\).
2. So \(c=0,\ a=-\dfrac12,\ b=\dfrac12\).
3. Coefficient of \(x^3\): \(-\dfrac16+\dfrac a6-\dfrac b6+\dfrac c3=-\dfrac13\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (c) \(e^{-2}\)

Explanation:
1. Let \(t=-x\to\infty\): \(f=\left(\dfrac t{t+2}\right)^t=\left(1+\dfrac2t\right)^{-t}\).
2. The limit is \(e^{-2}\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (d) \(-\dfrac e2\)

Explanation:
1. \((1+x)^{1/x}=e^{\frac{\ln(1+x)}x}=e\cdot e^{-x/2+x^2/3-\dots}\approx e\left(1-\dfrac x2\right)\).
2. So \(f(x)\to-\dfrac e2\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.

Answer: (a) \(e^2\)

Explanation:
1. Form \(1^\infty\): exponent \(\dfrac1{x^2}\cdot\dfrac{2x^2}{1+3x^2}\to2\).
2. The limit is \(e^2\).

Teacher's Note:
1. Identify the indeterminate form first, then use standard limits, series expansions or the exponential form for 1 to the power infinity.
2. Check left-hand and right-hand limits separately when greatest-integer, modulus or piecewise functions appear.