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**Illustration 1: Find the following limit(1984)****limnâ†’âˆž [1/(1-n ^{2}) + 2/(1-n^{2}) + â€¦ + n/(1-n^{2})]**

= lim

= lim

= lim

= -1/2

**Illustration 2: lim _{xâ†’0} sin (Ï€ cos^{2}x) / x^{2}equals(2001)**

**1. â€“Ï€****2. Î **

**3. Ï€/2**

**4. 1****Solution:** We need to compute the following expression

lim_{xâ†’0} sin (Ï€ cos^{2}x) / x^{2}

= lim_{xâ†’0} sin (Ï€ - Ï€sin^{2}x) / x^{2}

= lim_{xâ†’0} sin (Ï€ sin^{2}x) / Ï€ sin^{2}x .Ï€ sin^{2}x/Ï€x^{2} . Ï€

= 1.1.Ï€

=n

**Illustration 3: let f: Râ†’R besuch that f(1) = 3 and fâ€™(1) = 6. The find the value of lim _{xâ†’0 }[f(1+x)/ f(1)]^{1}^{/x}. (2002)**

So, log y = 1/x[log f(1+x) â€“ log f(1)]

So, lim

= fâ€™(1)/ f(1)

= 6/3

log (lim

lim

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