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lim_θ→0 [4θ (tan θ – 2θ tan θ)] / (1 – cos 2θ) =

1) 1/√2 2) 1/2 3) 1 4) 2 Solution: (4) 2 \(\begin{array}{l}\begin{array}{l} \lim _{\theta \rightarrow 0} \frac{4 \theta(\tan \theta-2 \theta... View Article

lim_θ→π/2 [(π / 2) – θ] / (cot θ) =

1) 0 2) -1 3) 1 4) ∞ Solution: (3) 1 \(\begin{array}{l}\begin{array}{l} \lim _{\theta \rightarrow \frac{\pi}{2}} { [(\pi / 2)... View Article

lim_x→0 (√1 – cos 2x) / (√2 x) =

1) 2 2) -1 3) 0 4) does not exist Solution: (4) does not exist \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}... View Article

lim_x→∞ [sin x / x] =

1) ∞ 2) 1 3) 0 4) does not exist Solution: (3) 0 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty} \frac{\sin x}{x}\\... View Article

lim_x→0 (cosec x)^{1/log x} =

1) 0 2) 1 3) 1/e 4) None of these Solution: (3) 1/e \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}(\operatorname{cosec} x)^{1 /... View Article

lim_x→0 [e^x – e^sinx] / [x – sin x] =

1) -1 2) 0 3) 1 4) None of these Solution: (3) 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}\left(\frac{e^{x}-e^{\sin x}}{x-\sin x}\right)... View Article

lim_x→0 [tan x – sin x] / x³ =

1) 1/2 2) – (1/2) 3) 0 4) 1 Solution: (1) ½ \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{\tan x-\sin x}{x^{3}}\\... View Article

For x > 0, lim_x→0 [(sin x)^1/x + (1 / x)^sinx] =

1) 0 2) -1 3) 1 4) 2 Solution: (3) 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}\left[(\sin x)^{1 / x}+\left(\frac{1}{x}\right)^{\sin x}\right]\\... View Article

The value of lim_ɑ→0 [cosec^-1 (sec ɑ) + cot^-1 (tan ɑ) + cot^-1 (sin^-1 ɑ)] / ɑ is

1) 0 2) -1 3) -2 4) 1 Solution: (3) -2 \(\begin{array}{l}\begin{array}{l} \lim _{\alpha \rightarrow 0} \frac{\operatorname{cosec}^{-1}(\sec \alpha)+\cot ^{-1}(\tan \alpha)+\cot... View Article

lim_x→0 (1 / x) sin^-1 [(2x) / (1 + x²)] =

1) -2 2) 0 3) 2 4) ∞ Solution: (3) 2 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{1}{x} \sin ^{-1}\left(\frac{2 x}{1+x^{2}}\right)... View Article

The value of lim_x→0 [∫₀^{x^2} sec² t dt] / [x sin x] =

1) 3 2) 2 3) 1 4) 0 Solution: (3) 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{\int_{0}^{x^{2}} \sec ^{2} t... View Article

The value of lim_x→∞ [(π / 2) – (tan^-1 x)]^1/x =

1) 0 2) 1 3) -1 4) e Solution: (2) 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty}\left(\frac{\pi}{2}-\tan ^{-1}x \right)^{1 / x}=l(\text... View Article

lim_x→0 (d / dx) [∫(1 – cos x) / x²] dx =

1) 1 2) 0 3) 1/2 4) None of these Solution: (3) ½ \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{d}{d x}... View Article

lim_x→1 [tan (x² – 1) ] / (x – 1) =

1) 2 2) 1/2 3) -2 4) – (1/2) Solution: (1) 2 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 1} \frac{\tan \left(x^{2}-1\right)}{x-1} \\... View Article

lim_x→0 [(1 – cos 2x) sin 5x] / [x² sin 3x] =

1) 10/3 2) 3/10 3) 6/5 4) 5/6 Solution: (1) 10/3 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{1-\cos 2 x \cdot... View Article

lim_x→0 x log_e (sin x) =

1) -1 2) loge 1 3) 1 4) None of these Solution: (2) loge 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}... View Article

The value of lim_x→0 [1 + sin x – cos x + log (1 – x)] / x³ is

1) -1 2) 1/2 3) -(1/2) 4) 1 Solution: (3) -(1/2) \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{1+\sin x-\cos x+\ln (1-x)}{x^{3}}... View Article

lim_x→0 [(1 – e^x) sin x] / (x² + x³) =

1) -1 2) 0 3) 1 4) 2 Solution: (1) -1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{\left(1-e^{x}\right) \sin x}{x^{2}+x^{3}} \\... View Article

The value of lim_x→2 [e^3x-6 – 1] / sin (2 – x) is

1) 3/2 2) 3 3) -3 4) -1 Solution: (3) -3 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 2} \frac{e^{3 x-6}-1}{\sin (2-x)} \quad... View Article

lim_x→π/2 [a^cotx – a^cosx] / [cot x – cos x], a > 0 =

1) loge π/2 2) loge 2 3) loge a 4) a Solution: (3) loge a \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \frac{\pi}{2}}... View Article

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