Limit Using Expansion - Practice Questions & MCQ

Limit Using Expansion (Part 1)

Using the expansions is one of the methods to find the limits. The following expansion formulas which are also known as Taylor series, are very useful in evaluating various limits.

 (i) $e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots \ldots$.
(ii) $a^x=1+x\left(\log _e a\right)+\frac{x^2}{2!}\left(\log _e a\right)^2+\frac{x^3}{3!}\left(\log _e a\right)^3+\ldots \ldots \ldots$
(iii) $\log (1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+\ldots \ldots \ldots$.
(iv) $\log (1-x)=-x-\frac{x^2}{2}-\frac{x^3}{3}-\frac{x^4}{4}-\ldots \ldots \ldots$.
(v) $(1+x)^n=1+n x+\frac{n(n-1)}{2!} x^2+\frac{n(n-1)(n-2)}{3!} x^3+\ldots \ldots \ldots$.

Limit Using Expansion (Part 2) (vi) $\quad \sin x=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\ldots \ldots \ldots$ (vii) $\quad \cos x=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\ldots \ldots$. (viii) $\quad \tan x=x+\frac{x^3}{3}+\frac{2}{15} x^5+\ldots \ldots$. (ix) $\sin ^{-1} x=x+\frac{1^2}{3!} \cdot x^3+\frac{1^2 \cdot 3^2}{5!} \cdot x^5+\frac{1^2 \cdot 3^2 \cdot 5^2}{7!} \cdot x^7 \ldots \ldots \ldots$ (x) $\tan ^{-1} x=x-\frac{x^3}{3}+\frac{x^5}{5}+\ldots \ldots \ldots$.