LIMITS (Revision)

\[1.\ Evaluate:\ (i)\ \lim\ _{x\ \to\ 3}\ \frac{x^6\ -\ 3^6}{x\ -\ 3}\ \hspace{1cm}\ (ii)\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 7\ θ}{Sin\ 2\ θ}\ \hspace{7cm}\]
\[\color {black}{Solution:}\ \hspace{17cm}\]
\[(i)\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{10cm}\]
\[\lim\ _{x\ \to\ 3}\ \frac{x^6\ -\ 3^6}{x\ -\ 3}\ =\ 6\ 3^{6\ -\ 1}\ =\ 6\ 3^5\ =\ 6(243)\ =\ 1458\ \hspace{10cm}\]
\[(ii)\ W.\ K.\ T\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ n\ θ}{θ}\ =\ n\ \hspace{10cm}\]
\[Multiply\ and\ divide\ by\ ‘\ θ\ ‘\]
\[\lim\ _{θ\ \to\ 0}\ \frac{\frac{Sin\ 7\ θ}{θ}}{\frac{Sin\ 2\ θ}{θ}}\ \hspace{10cm}\]
\[=\ \frac{7}{2}\ \hspace{10cm}\]
\[2:\ Evaluate:\ \lim\ _{x\ \to\ 3}\ \frac{x^4\ -\ 81}{x^2\ -\ 9}\ \hspace{15cm}\]
\[\color {black}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}\]
\[\lim\ _{x\ \to\ 3}\ \frac{x^4\ -\ 81}{x^2\ -\ 9}\ =\ \lim\ _{x\ \to\ 3}\ \frac{x^4\ -\ 3^4}{x^2\ -\ 3^2} \hspace{10cm}\]
\[Multiply\ and\ divide\ by\ ‘x – 3’\]
\[=\ \lim\ _{x\ \to\ 3}\ \frac{\frac{x^4\ -\ 3^4}{x\ -\ 3}}{\frac{x^2\ -\ 3^2}{x\ -\ 3}}\ \hspace{10cm}\]
\[=\ \frac{4\ (3^{4\ -\ 1})}{2\ (3^{2\ -\ 1})}\ \hspace{10cm}\]
\[=\ \frac{2\ 3^3}{3^1}\ \hspace{10cm}\]
\[=\ 2\ 3^{3\ -\ 1}\ \hspace{10cm}\]
\[=\ 2\ 3^{2}\ \hspace{10cm}\]
\[=\ 18\ \hspace{10cm}\]