LIMITS (Excercise)

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

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