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If lim_x→0 [(e^kx – 1) sin kx] / x² = 4, then k =

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

The value of lim_x→0 [1 – cos (1 – cos x)] / x⁴ is

1) 1/2 2) 1/4 3) 1/6 4) 1/8 Solution: (4) ⅛ \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^{4}} \\... View Article

The value of lim_x→0 [sin² x + cos x – 1] / x² =

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

lim_x→0 [sin |x|] / x =

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

The value of lim_x→π/2 [sin (cos x) (cos x)] / [sin x – cosec x] is

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

lim_x→2 [(√1 – {cos 2 (x – 2)}) / (x – 2)] =

1) √2 2) -√2 3) 1/√2 4) does not exist Solution: (4) does not exist \(\begin{array}{l}\begin{array}{l} \text { Let }... View Article

Let f (x) = ∫₁^x sin t / (t) dt, lim_x→∞ f’ (x) =

1) 0 2) 1 3) 2 4) None of these Solution: (1) 0 \(\begin{array}{l}\begin{array}{l} \operatorname{lim}_{x \rightarrow \infty} f'(x)=\lim _{x \rightarrow... View Article

lim_x→0 [(2 + x) sin (2 + x) – 2 sin 2] / x =

1) sin 2 2) cos 2 3) 1 4) 2 cos 2 + sin 2 Solution: (4) 2 cos 2... View Article

lim_x→0 [(1 + tan x) / (1 + sin x)]^x =

1) 1/e 2) 1 3) e 4) e2 5) None of these Solution: (2) 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}\left(\frac{1+\tan... View Article

lim_x→0 [(1 – cos 2x) (3 + cos x)] / [x tan 4x] =

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

lim_x→2 [sin^-1 (x – 2)] / [x² – 4] =

1) 0 2) 2 3) 1/4 4) 1 Solution: (3) ¼ \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 2} \frac{\sin ^{-1}(x-2)}{x^{2}-4} \\ \Rightarrow... View Article

The value of lim_x→0 [∫₀^x cos t dt] / x =

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

If lim_x→0 [2a sin x – sin 2x] / [tan³ x] exists and is equal to 1, then the value of a is

1) 2 2) 1 3) 0 4) -1 Solution: (2) 1 \(\begin{array}{l}\lim _{\mathrm{x} \rightarrow 0} \frac{2 \mathrm{a} \sin \mathrm{x}-\sin 2... View Article

lim_x→0 [e^sinx – 1] / x =

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

lim_x→π/2 [(1 – tan (x / 2) (1 – sin x)] / [(1 + tan (x / 2) (π – 2x)³] =

1) 1/8 2) 0 3) 1/32 4) ∞ Solution: (3) 1/32 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \frac{\pi}{2}}\left(\frac{1-\tan x / 2}{1+\tan x... View Article

lim_{x→tan^(-1) 3} [(tan² x – 2 tan x – 3) / (tan² x – 4 tan x + 3)] =

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

lim_x→0 [sin (π cos² x) / x²] =

1) π/2 2) 1 3) – π 4) π Solution: (4) π \(\begin{array}{l}\lim _{x \rightarrow 0} \frac{\sin \left(\pi \cos ^{2}... View Article

For x belongs to R, lim_x→∞ [(x – 3) / (x + 2)]^x =

1) e 2) e-1 3) e-5 4) e5 Solution: (3) e-5 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty}\left(\frac{x-3}{x+2}\right)^{x} \\ \Rightarrow \lim _{x... View Article

lim_x→0 (1 – ax)^1/x =

1) e-a 2) e 3) ea 4) 1 Solution: (1) e-a \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0}(1-a x)^{\frac{1}{x}}\\ \Rightarrow e_{x \rightarrow... View Article

If f(a) = 2, f ‘ (a) = 1, g(a) = 3, g ‘(a) = -1 then lim_x→a [f (a) g (x) – f (x) g (a)] / (x – a) =

1) 6 2) 1 3) -1 4) -5 Solution: (4) -5 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow a} \frac{f(a) g(x)-f(x) g(a)}{x-a}\\ \text... View Article

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