Question

find the value of following limits​

Answer 1

$\sf\red{\underline{\underline{Question}}}:$

• $\displaystyle\lim_{ x \to -1} [ 1 + x + x^2 + .... + x^{10} ]$

$\sf\red{\underline{\underline{Solution}}}:$

$\displaystyle\lim_{ x \to -1} [ 1 + x + x^2 + ... + x^{10} ]$

Here we can see 1 + x + x² + ...+ x¹⁰ is in form of GP. so we can use here Sum of GP formula .

$\boxed{\red\bigstar\bf Sum\; of\; GP (S_n) = \dfrac{a(r^n -1)}{r-1}}$

where:

• a = the first term of GP.
• r = the common ratio of GP.
• n = nth term of GP.

here we have,

• a = 1
• r = 1/x
• n = 10

putting all values in formula :

$\sf\displaystyle \lim_{x \to -1}\left[\dfrac{1\left(\left(\frac{1}{x}\right)^{10}-1\right)}{\frac{1}{x}-1}\right]$

$\rm\displaystyle \lim_{x \to -1}\left[\dfrac{\dfrac{1}{x^{10}}-1}{\dfrac{1}{x}-1}\right]$

Substituting limits :

$= \dfrac{\dfrac{1}{(-1)^{10}}-1}{\dfrac{1}{-1}-1}$

$\rm=\dfrac{\dfrac{1}{1}-1}{\dfrac{1}{-1}-1}$

$\rm =\dfrac{1-1}{-1-1}$

$\rm=\dfrac{0}{2}$

$= 0$

$\sf\red{\underline{\underline{Therefore}}},$

• $\displaystyle\lim_{ x \to -1} [ 1 + x + x^2 + .... + x^{10} ] = 0$