JEE Mathematics Limits Indeterminate Forms MCQs Set 06

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Multiple Choice Questions

Question 1: If f(x) and g(x) are differentiable functions and f(1) = g(1) = 2, then \( \lim_{x\to1}\frac{f(1)g(x)-f(x)g(1)-f(1)+g(1)}{g(x)-f(x)} \) is equal to
(a) 0
(b) 1
(c) 2
(d) none of these
Show Answer & Explanation

Answer: (c) 2

Explanation:
1. With f(1) = g(1) = 2, the numerator is 2g(x) − 2f(x) − 2 + 2 = 2(g(x) − f(x)).
2. So the ratio is 2 near x = 1 (where g(x) ≠ f(x)).

Teacher's Note:
1. Substitute the given values before taking the limit.

Question 2: If f(a) = 2, f′(a) = 1, g(a) = −1, g′(a) = 2, then \( \lim_{x\to a}\frac{g(x)f(a)-g(a)f(x)}{x-a} \) is
(a) −5
(b) \( \frac15 \)
(c) 5
(d) none of these
Show Answer & Explanation

Answer: (c) 5

Explanation:
1. The numerator vanishes at x = a (g(a)f(a) − g(a)f(a) = 0).
2. By L'Hôpital: g′(a)f(a) − g(a)f′(a) = 2·2 − (−1)(1) = 5.

Teacher's Note:
1. The expression is the derivative of g(x)f(a) − g(a)f(x) at x = a.

Question 3: \( \lim_{n\to\infty}\left(1+\sin\frac an\right)^n \) is equal to
(a) e^(a/2)
(b) eᵃ
(c) e
(d) e^(2a)
Show Answer & Explanation

Answer: (b) eᵃ

Explanation:
1. This is of the form 1^∞: limit = exp(lim n sin(a/n)) = exp(a).
2. Since n sin(a/n) → a.

Teacher's Note:
1. For 1^∞ use exp(lim g(x)(f(x) − 1)).

Question 4: \( \lim_{x\to0}\left\{\tan\left(\frac\pi4-x\right)\right\}^{1/x} \) is equal to
(a) 1
(b) e
(c) e²
(d) e⁻²
Show Answer & Explanation

Answer: (d) e⁻²

Explanation:
1. tan(π/4 − x) = (1 − tan x)/(1 + tan x) ≈ 1 − 2x.
2. Limit = exp(lim (−2x)/x) = e⁻².

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 5: \( \lim_{x\to\pi/4}(2-\tan x)^{\log\tan x} \) is equal to
(a) 0
(b) 1
(c) e
(d) e⁻¹
Show Answer & Explanation

Answer: (b) 1

Explanation:
1. As x → π/4, tan x → 1, so 2 − tan x → 1 and log tan x → 0.
2. The form is 1⁰, with limit exp(lim ln(tan x)·ln(2 − tan x)) = e⁰ = 1.

Teacher's Note:
1. Take logarithms: the exponent is a product of two quantities tending to 0.

Question 6: \( \lim_{x\to\infty}\left(\frac{x-1}{x+1}\right)^{x+2} \) is equal to
(a) e
(b) e⁻¹
(c) e⁻²
(d) none of these
Show Answer & Explanation

Answer: (c) e⁻²

Explanation:
1. ((x − 1)/(x + 1))^(x+2) = (1 − 2/(x + 1))^(x+2).
2. The limit is exp(lim −2(x + 2)/(x + 1)) = e⁻².

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 7: If \( \lim_{x\to\infty}\left(1+\frac\lambda x+\frac{\mu}{x^2}\right)^{2x}=e^2 \), then
(a) λ = 1, μ = 2
(b) λ = 2, μ = 1
(c) λ = 1, μ = any real constant
(d) λ = μ = 1
Show Answer & Explanation

Answer: (c) λ = 1, μ = any real constant

Explanation:
1. The limit equals exp(lim 2x(λ/x + μ/x²)) = e^(2λ).
2. e^(2λ) = e² gives λ = 1, and μ plays no role.

Teacher's Note:
1. Identify the form of the limit first.
2. Use series or standard limits to finish.

Question 8: \( \lim_{\theta\to0^+}\frac{\sin\sqrt\theta}{\sqrt{\sin\theta}} \) is equal to
(a) 0
(b) 1
(c) −1
(d) none of these
Show Answer & Explanation

Answer: (b) 1

Explanation:
1. sin√θ ≈ √θ and √(sin θ) ≈ √θ.
2. The ratio tends to 1.

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 9: \( \lim_{x\to\infty}\frac{\log_e[x]}{x},\text{ where }[\cdot]\text{ is the greatest integer function} \) is equal to
(a) 0
(b) 1
(c) −1
(d) non-existent
Show Answer & Explanation

Answer: (a) 0

Explanation:
1. [x] ≤ x, so 0 ≤ ln[x]/x ≤ ln x/x for x ≥ 1.
2. ln x/x → 0, so the limit is 0.

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 10: \( \lim_{x\to1}\frac{x\sin\{x-[x]\}}{x-1} \), where [·] is the greatest integer function, is
(a) 0
(b) −1
(c) not existent
(d) none of these
Show Answer & Explanation

Answer: (c) not existent

Explanation:
1. For x → 1⁺, x − [x] = x − 1 and the quotient → 1.
2. For x → 1⁻, x − [x] = x, so x sin x/(x − 1) → −∞.
3. The limit does not exist.

Teacher's Note:
1. Use [x] = 0 just left of 1 and [x] = 1 just right of 1.

Question 11: \( \lim_{x\to2}\{[2-x]+[x-2]-x\} \) is equal to
(a) is 0
(b) is 3
(c) is −3
(d) does not exist
Show Answer & Explanation

Answer: (c) is −3

Explanation:
1. For x → 2⁺: [2 − x] = −1, [x − 2] = 0, giving −1 − x → −3.
2. For x → 2⁻: [2 − x] = 0, [x − 2] = −1, giving −1 − x → −3.
3. Both sides give −3.

Teacher's Note:
1. Check both sides because of the greatest integer function.

Question 12: \( \lim_{x\to-1}\{[x]+|x|\} \) is equal to
(a) is 0
(b) is 1
(c) does not exist
(d) none of these
Show Answer & Explanation

Answer: (c) does not exist

Explanation:
1. For x → −1⁺: [x] = −1, |x| → 1, giving 0.
2. For x → −1⁻: [x] = −2, |x| → 1, giving −1.
3. The limits differ, so the limit does not exist.

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 13: Let f(x) = x² − 1 for 0 < x < 2 and f(x) = 2x + 3 for 2 ≤ x < 3. The quadratic equation whose roots are \( \lim_{x\to2^-}f(x) \) and \( \lim_{x\to2^+}f(x) \) is
(a) x² − 6x + 9 = 0
(b) x² − 10x + 21 = 0
(c) x² − 14x + 49 = 0
(d) none of these
Show Answer & Explanation

Answer: (b) x² − 10x + 21 = 0

Explanation:
1. Left limit: 4 − 1 = 3. Right limit: 2·2 + 3 = 7.
2. Sum 10 and product 21, so x² − 10x + 21 = 0.

Teacher's Note:
1. Identify the form of the limit first.
2. Use series or standard limits to finish.

Question 14: If [·] is the greatest integer function, then \( \lim_{n\to\infty}\frac{[x]+[2x]+\cdots+[nx]}{n^2} \) is
(a) 0
(b) x
(c) \( \frac x2 \)
(d) \( \frac{x^2}{2} \)
Show Answer & Explanation

Answer: (c) \( \frac x2 \)

Explanation:
1. kx − 1 < [kx] ≤ kx, so the sum lies between x·n(n + 1)/2 − n and x·n(n + 1)/2.
2. Dividing by n², both bounds tend to x/2.

Teacher's Note:
1. Squeeze using [y] ≤ y < [y] + 1.

Question 15: \( \lim_{x\to0}\frac{\int_0^{x^2}\cos t^2\,dt}{x\sin x} \) is equal to
(a) 1
(b) 2
(c) 0
(d) none of these
Show Answer & Explanation

Answer: (a) 1

Explanation:
1. Apply L'Hôpital: numerator derivative 2x cos x⁴; denominator derivative sin x + x cos x.
2. Dividing by x: 2cos x⁴/(sin x/x + cos x) → 2/2 = 1.

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 16: \( \lim_{x\to a}\frac{x}{x-a}\int_a^xf(t)\,dt \) is equal to
(a) f(a)
(b) a f(a)
(c) 0
(d) none of these
Show Answer & Explanation

Answer: (b) a f(a)

Explanation:
1. (1/(x − a))∫ₐˣ f → f(a) (derivative of the integral at a).
2. Multiplying by x → a gives a f(a).

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 17: \( \lim_{x\to1^+}\frac{\int_1^x|t-1|\,dt}{\sin(x-1)} \) is equal to
(a) 0
(b) 1
(c) −1
(d) none of these
Show Answer & Explanation

Answer: (a) 0

Explanation:
1. For x > 1 the integral is (x − 1)²/2.
2. The ratio (x − 1)²/(2 sin(x − 1)) ≈ (x − 1)/2 → 0.

Teacher's Note:
1. Identify the indeterminate form first.
2. Apply standard limits or expansions.

Question 18: If f(x) is continuous in [0, 1] and f(1/3) = 1, then \( \lim_{n\to\infty}f\left(\frac{n}{\sqrt{9n^2+1}}\right) \) is
(a) 1
(b) 0
(c) \( \frac13 \)
(d) none of these
Show Answer & Explanation

Answer: (a) 1

Explanation:
1. n/√(9n² + 1) → 1/3.
2. By continuity the limit is f(1/3) = 1.

Teacher's Note:
1. Identify the form of the limit first.
2. Use series or standard limits to finish.

Question 19: If f(x) is continuous and f(9/2) = 2/9, then \( \lim_{x\to0}f\left(\frac{1-\cos3x}{x^2}\right) \) is equal to
(a) \( \frac92 \)
(b) \( \frac29 \)
(c) 0
(d) none of these
Show Answer & Explanation

Answer: (b) \( \frac29 \)

Explanation:
1. (1 − cos 3x)/x² → 9/2.
2. By continuity the limit is f(9/2) = 2/9.

Teacher's Note:
1. Identify the form of the limit first.
2. Use series or standard limits to finish.

Question 20: If α is a repeated root of ax² + bx + c = 0, then \( \lim_{x\to\alpha}\frac{\sin(ax^2+bx+c)}{(x-\alpha)^2} \) is
(a) 0
(b) a
(c) b
(d) c
Show Answer & Explanation

Answer: (b) a

Explanation:
1. A repeated root gives ax² + bx + c = a(x − α)².
2. sin(a(x − α)²)/(x − α)² → a.

Teacher's Note:
1. Identify the form of the limit first.
2. Use series or standard limits to finish.

Question 21: If f(x) = |x − 1| − [x], where [x] is the greatest integer less than or equal to x, consider the statements:
1. f(1 + 0) = −1 and f(1 − 0) = 0
2. f(1 + 0) = 0 = −f(1 − 0)
3. the limit of f(x) as x → 1 exists
4. the limit of f(x) as x → 1 does not exist
(a) 1 and 4 only
(b) 1, 2 and 4 only
(c) All of 1, 2, 3 and 4
(d) 1, 3 and 4 only
Show Answer & Explanation

Answer: (a) 1 and 4 only

Explanation:
1. For x slightly above 1: |x − 1| → 0 and [x] = 1, so f(1 + 0) = −1.
2. For x slightly below 1: |x − 1| → 0 and [x] = 0, so f(1 − 0) = 0.
3. The one-sided limits differ, so statements 1 and 4 hold.

Teacher's Note:
1. Find the one-sided limits using the value of [x] on each side.

Question 22: If \( \lim_{n\to\infty}\left(an-\frac{1+n^2}{1+n}\right)=b \), a finite number, consider the statements:
1. a = 1
2. a = 0
3. b = 1
4. b = −1
(a) 1, 2 and 3 only
(b) All of 1, 2, 3 and 4
(c) 1 and 3 only
(d) 1, 3 and 4 only
Show Answer & Explanation

Answer: (c) 1 and 3 only

Explanation:
1. (1 + n²)/(1 + n) = n − 1 + 2/(1 + n).
2. For finiteness a = 1, and then an − (n − 1 + 2/(n + 1)) → 1.
3. So a = 1 and b = 1.

Teacher's Note:
1. Do polynomial division to see the leading behaviour.

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