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nunknown91

For the functions of the set

{ $x, sinx, tanx, {sin}^{-1}x, {tan}^{-1}x, sinhx, tanhx, {sinh}^{-1}x, {tanh}^{-1}x$ }

1. $\mathop{\lim}\limits_{{x} \to {0}} \frac{any function}{any function} = 1$

e.g.

1. $\mathop{\lim}\limits_{{x} \to {0}}\frac{sinx}{sinhx} =1$

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2. $\mathop{\lim}\limits_{{x} \to {0}} \frac{f(nx)}{g(mx)} = \frac{n}{m}$

e.g

1 . $\mathop{\lim}\limits_{{x} \to {0}} \frac{tan3x}{tan2x} = \frac{3}{2}$

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3. $\mathop{\lim}\limits_{{x} \to {0}} \frac{f(nx) \pm g(mx)}{h(px) \pm j(qx)} = \frac{n \pm m}{p \pm q}$

e.g.

1. $\mathop{\lim}\limits_{{x} \to {0}} \frac{sin3x + tan2x}{tanh4x - {sin}^{-1}2x} = \frac{3+2}{4-2} = \frac{5}{2}$

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4. $\mathop{\lim}\limits_{{x} \to {0}} \frac{{f}^{a}(mx) . {g}^{b}(nx)}{{h}^{c}(px) . {j}^{d}(qx)} = \frac{{m}^{a} . {n}^{b}}{{p}^{c} . {q}^{d}} if a+b = c+d$

e.g.

1. $\mathop{\lim}\limits_{{x} \to {0}} \frac{{sin}^{2}(2x) . {sinh}^{3}(x)}{tan3x. {tanh}^{4}(5x)} = \frac{{2}^{2} . {1}^{3}}{{3}^{1} . {5}^{4}} = \frac{8}{1875}$

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5. $\mathop{\lim}\limits_{{x} \to {0}} \frac{1 - cosax}{f(mx)} = 0$

e.g.

1. $\mathop{\lim}\limits_{{x} \to {0}} \frac{1 - cos3x}{tan4x} = 0$

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6. $\mathop{\lim}\limits_{{x} \to {0}} \frac{1 - {cos}^{p}ax}{f(mx).g(nx)} = \frac{p . {a}^{2}}{2mn}$

e.g.

1. $\mathop{\lim}\limits_{{x} \to {0}} \frac{1 - {cos}^{3}2x}{sinhx . tanh(4x)} = \frac{3 . {2}^{2}}{2.1.4} = \frac{3}{2}$

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7. $\mathop{\lim}\limits_{{x} \to {0}} \frac{cosax - cosbx}{f(mx) . g(nx)} = \frac{{b}^{2} - {a}^{2}}{2mn}$

e.g.

1. $\mathop{\lim}\limits_{{x} \to {0}} \frac{cosx - cos4x}{sinh(2x) . {tanh}^{-1}(3x)} = \frac{{4}^{2} - {1}^{2}}{2.2.3} = \frac{5}{4}$

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I hope these formulae will be useful in saving the time in competitive exams.

Although it is tough to remember some of them but few are very easy.

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