Question

Find the limits :
$i.\underset{x\to 1}{lim}(x^3-x^2+1)$
$ii.\underset{x\to 3}{lim}\left[x\right(x+1\left)\right]$
$iii.\underset{x\to -1}{lim}(1+x+x^2+...+x^{10})$

Answer 1

$\left(i\right)$ Given: $\underset{x\to 1}{lim}(x^3-x^2+1)$
Substituing $x=1$
$=(1{)}^3-(1{)}^2+1$
$=1-1+1$
$=1$
$\therefore \underset{x\to 1}{lim}[x^3-x^2+1]=1$

$\left(ii\right)$ Given: $\underset{x\to 3}{lim}\left[x\right(x+1\left)\right]$
Substituing $x=3$
$=3(3+1)$
$=3\left(4\right)$
$=12$
$\therefore \underset{x\to 3}{lim}\left[x\right(x+1\left)\right]=12$

$\left(iii\right)$ Given: $\underset{x\to -1}{lim}(1+x+x^2+...+x^{10})$
Substituing $x=-1$
$=1+(-1)+(-1{)}^2+(-1{)}^3+(-1{)}^4+(-1{)}^5+(-1{)}^6+(-1{)}^7+(-1{)}^8+(-1{)}^9+(-1{)}^{10}$
$=1+(-1+1)+(-1+1)+(-1+1)+(-1+1)+(-1+1)$
$=1+0+0+0+0+0$
$=1$
$\therefore \underset{x\to -1}{lim}(1+x+x^2+...+x^{10})=1$