GATE (TF) Textile 2019 Question Paper Solution | GATE/2019/TF/01

Question 01 (Textile Technology & Fibre Science)

The value of \lim_{x\rightarrow 0}\frac{e^{x}-1-x}{x^{2}} is

(A)-\frac{1}{2}
(B)0
(C)\frac{1}{2}
(D)1
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\lim_{x\rightarrow 0}\frac{e^{x}-1-x}{x^{2}}

As we know the equation :

\Rightarrow e^x=1+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+………..+\frac{x^n}{n}

By putting the equation of e^x we get :

\lim_{x\rightarrow 0}\frac{1+x+\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+………..+\frac{x^n}{n}-1-x}{x^{2}}

\lim_{x\rightarrow 0}\frac{\frac{x^2}{2}+\frac{x^3}{3}+\frac{x^4}{4}+………..+\frac{x^n}{n}}{x^{2}}

\lim_{x\rightarrow 0}\frac{\frac{1}{2}+\frac{x}{3}+\frac{x^2}{4}+………..+\frac{x^{n-2}}{n}}

By taking limit-

=\frac{1}{2} (Ans)

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