# LIMITS (Excercise)

$\LARGE{\color {purple} {PART- A}}$
$\color {purple} {1.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 2}\ \frac{x^2\ -\ 2^2}{x\ -\ 2}\ \hspace{15cm}$
$\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}$
$\lim\ _{x\ \to\ 2}\ \frac{x^2\ -\ a^2}{x\ -\ 2}\ =\ 2\ 2^{2\ -\ 1}\ =\ 2\ 2^1\ =\ 2(2)\ =\ 4\ \hspace{10cm}$
$\LARGE{\color {purple} {PART- B}}$
$\color {purple} {2.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ a}\ \frac{\sqrt{x}\ -\ \sqrt{a}}{x\ -\ a}\ \hspace{15cm}$
$\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}$
$\lim\ _{x\ \to\ a}\ \frac{\sqrt{x}\ -\ \sqrt{a}}{x\ -\ a}\ =\ \lim\ _{x\ \to\ a}\ \frac{x^ {\frac{1}{2}}\ -\ a^{\frac{1}{2}}}{x\ -\ a}\ \hspace{10cm}$
$=\ \frac{1}{2}\ a^{\frac{1}{2}\ -\ 1}\ =\ \frac{1}{2}\ a^{\frac{-1}{2}}\ \hspace{10cm}$
$\color {purple} {3.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 7x}{Sin\ 3x}\ \hspace{15cm}$
$\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ nx}{x}\ =\ n\ \hspace{15cm}$
$Multiply\ and\ divide\ by\ ‘x ‘$
$\lim\ _{x\ \to\ 0}\ \frac{\frac{Sin\ 7x}{x}}{\frac{Sin\ 3x}{x}}\ \hspace{10cm}$
$=\ \frac{7}{3}\ \hspace{10cm}$
$\color {purple} {4.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 2x}{Sin\ 4x}\ \hspace{15cm}$
$\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ nθ}{θ}\ =\ n\ \hspace{15cm}$
$Multiply\ and\ divide\ by\ ‘x ‘$
$\lim\ _{x\ \to\ 0}\ \frac{\frac{Sin\ 2x}{x}}{\frac{Sin\ 4x}{x}}\ \hspace{10cm}$
$=\ \frac{2}{4}\ \hspace{10cm}$
$=\ \frac{1}{2}\ \hspace{10cm}$
$\LARGE{\color {purple} {PART- C}}$
$\color {purple} {5.}\ \color {red} {Evaluate:}\ (i)\ \lim\ _{x\ \to\ 0}\ \frac{x^2\ +\ 7\ x}{x^2\ +\ 5\ x}\ \hspace{1cm}\ (ii)\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 3\ θ}{Sin\ 2\ θ}\ \hspace{7cm}$
$\color {blue}{Solution:}\ \hspace{17cm}$
$(i)\ \lim\ _{x\ \to\ 0}\ \frac{x^2\ +\ 7\ x}{x^2\ +\ 5\ x}\ \hspace{10cm}$
$=\ \lim\ _{x\ \to\ 0}\ \frac{x(x\ +\ 7)}{x(x\ +\ 5)}\ \hspace{10cm}$
$=\ \frac{(0\ +\ 7)}{(0\ +\ 5)}\ \hspace{10cm}$
$=\ \frac{7}{5}\ \hspace{10cm}$
$(ii)\ W.\ K.\ T\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ n\ θ}{θ}\ =\ n\ \hspace{10cm}$
$Multiply\ and\ divide\ by\ ‘\ θ\ ‘$
$\lim\ _{θ\ \to\ 0}\ \frac{\frac{Sin\ 3\ θ}{θ}}{\frac{Sin\ 2\ θ}{θ}}\ \hspace{10cm}$
$=\ \frac{3}{2}\ \hspace{10cm}$
$\color {purple} {6.}\ \color {red} {Evaluate:}\ (i)\ \lim\ _{x\ \to\ 3}\ \frac{x^6\ -\ 3^6}{x\ -\ 3}\ \hspace{1cm}\ (ii)\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 7\ θ}{Sin\ 2\ θ}\ \hspace{7cm}$
$\color {black}{Solution:}\ Question\ 1\ \hspace{17cm}$

https://yanamtakshashila.com/2021/12/23/limits-revision/

$\color {purple} {7.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 3}\ \frac{x^5\ -\ 3^5}{x^4\ -\ 3^4}\ \hspace{15cm}$
$\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}$
$Multiply\ and\ divide\ by\ ‘x – 3’$
$=\ \lim\ _{x\ \to\ 3}\ \frac{\frac{x^5\ -\ 3^5}{x\ -\ 3}}{\frac{x^4\ -\ 3^4}{x\ -\ 3}}\ \hspace{10cm}$
$=\ \frac{5\ (3^{5\ -\ 1})}{4\ (3^{4\ -\ 1})}\ \hspace{10cm}$
$=\ \frac{5\ 3^4}{4\ 3^3}\ \hspace{10cm}$
$=\ \frac{5}{4}\ 3^{4\ -\ 3}\ \hspace{10cm}$
$=\ \frac{5}{4}\ 3^1\ \hspace{10cm}$
$=\ \frac{15}{4}\ \hspace{10cm}$