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lim_x→∞ [√x² + 2x – 1 – x] =

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

lim_x→∞ [1 – (4 / (x – 1))]^3x-1 =

1) e12 2) e-12 3) e4 4) e3 Solution: (2) e-12 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty}\left(1-\frac{4}{x-1}\right)^{(3 x-1)}\\ \Rightarrow e^{\lim _{x... View Article

If f(2) = 4 and f'(2) = 4. Then, lim_x→2 [x f (2) – 2 f (x)] / (x – 2) =

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

lim_x→∞ [(x² + 5x + 3) / (x² + x + 2)]^x =

1) e4 2) e2 3) e3 4) e Solution: (1) e4 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty}\left(\frac{x^{2}+5 x+3}{x^{2}+x+2}\right)^{x} \\ \Rightarrow \lim... View Article

If f (x) = {1, when x is rational, 0 when x is irrational is, then lim_x→0 f (x) =

1) 0 2) 1 3) 1/2 4) None of these Solution: (4) None of these \(\begin{array}{l}f(x)=\left\{\begin{array}{ll}1, & x \in q... View Article

lim_x→0 [a^x + a^-x – 2] / x² =

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

lim_n→∞ [2^-n (n² + 5n + 6)] / (n + 4) (n + 5) =

1) 0 2) 1 3) ∞ 4) – ∞ Solution: (1) 0 \(\begin{array}{l}\begin{array}{l} \lim _{n \rightarrow \infty} \frac{2^{-n}\left(n^{2}+5 n+6\right)}{(n+4)(n+5)}\\ \Rightarrow... View Article

lim_x→1 [e^-x – e^-1] / (x – 1) =

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

lim_x→∞ [(2x – 3) (3x – 4)] / [(4x – 5) (5x – 6)] =

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

The value of lim_x→∞ [(x² + bx + 4) / (x² + ax + 5)] is

1) b/a 2) 0 3) 1 4) 4/5 Solution: (3) 1 \(\begin{array}{l}\lim _{x \rightarrow \infty}\left(\frac{x^{2}+b x+4}{x^{2}+a x+5}\right)\\ \text { Divide... View Article

lim_x→0 [a^x – b^x] / (e^x – 1) =

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

The value of lim_x→∞ √a²x² + ax + 1 – √a²x² + 1 is

1) 1/2 2) 1 3) 2 4) None of these Solution: (1) ½ \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \infty} \sqrt{a^{2} x^{2}+a... View Article

If f : R → R be a differentiable function having f (2) = 6, f'(2) = (1/48). Then lim_x→2 [∫₆^{f (x)} 4t³ dt] / [x – 2] =

1) 18 2) 12 3) 36 4) 24 Solution: (1) 18 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 2} \frac{\int_{6}^{f(x)} 4 t^{3} d... View Article

The value of lim_x→∞ [(x² – 2x + 1) / (x² – 4x + 2)]^x is

1) e2 2) e-2 3) e6 4) None of these Solution: (1) e2 \(\begin{array}{l} \lim _{x \rightarrow \infty}\left(\frac{x^{2}-2 x+1}{x^{2}-4 x+2}\right)^{x}\left(\infty^{\infty}... View Article

If f(x) differentiable and f” (0) = a, then lim_x→0 [2 f (x) – 3 f (2x) + f (4x)] / x² =

1) 3a 2) 2a 3) 5a 4) 4a Solution: (1) 3a \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{2 f(x)-3 f(2 x)+f(4... View Article

If lim_x→∞ [(x³ + 1) / (x² + 1)] – (ax + b) = 2, then

1) a = 1 and b = 1 2) a = 1 and b = – 1 3) a =... View Article

lim_x→π/4 [∫₂^{sec^2x} f (t) dt] / [x² – (π² / 16)] =

1) 8/π f(2) 2) 2/πf(2) 3) 2/πf(1/2) 4) 4f(2) Solution: (1) 8/π f(2) \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow \frac{\pi}{4}} \frac{\int_{2}^{\sec ^{2}... View Article

Let f : R → R be a differentiable function and f (1) = 4, Then the value of lim_x→1 ∫₄^{f (x)} 2t / (x – 1) dt is

1) 8 f ‘(1) 2) 4 f ‘(1) 3) 2 f ‘(1) 4) f ‘(1) Solution: (1) 8 f ‘(1)... View Article

If 0 < p < q, then lim_n→∞ (qⁿ + pⁿ)^1/n =

1) e 2) p 3) q 4) 0 Solution: (3) q \(\begin{array}{l}\begin{array}{l} \lim _{n \rightarrow \infty}\left(q^{n}+p^{n}\right)^{\frac{1}{n}} \text { as }... View Article

lim_x→0 [e^5x – e^4x] / x =

1) 1 2) 2 3) 4 4) 5 Solution: (1) 1 \(\begin{array}{l}\begin{array}{l} \lim _{x \rightarrow 0} \frac{e^{5 x}-e^{4 x}}{x} \\... View Article

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