\[\color {royalblue} {Definition}:\ \hspace{20cm}\]
\[When\ the\ variable\ x\ approaches\ a\ constant\ and\ if\ the\ function\ f ( x )\ approaches\ a\ constant\ l,\]
\[then\ l\ is\ called\ the\ limit\ value\ of\ f ( x )\ as\ x\ approaches\ a\ and\ is\ denoted\ as\ \lim\ _{x\ \to\ a}\ f(x)\ =\ l.\]
\[\color {royalblue} {Properties}:\ \hspace{20cm}\]
\[1)\ \lim\ _{x\ \to\ a}\ [f(x)\ \pm\ g(x)]\ =\ lim\ _{x\ \to\ a}\ f(x)\ \pm\ lm\ _{x\ \to\ a}\ g(x)\]
\[2)\ \lim\ _{x\ \to\ a}\ [K\ f(x)\ ]\ =\ K\ \lim\ _{x\ \to\ a}\ f(x)\]
\[3)\ \lim\ _{x\ \to\ a}\ [f(x)\ .\ g(x)]\ =\ lim\ _{x\ \to\ a}\ f(x)\ . \ lm\ _{x\ \to\ a}\ g(x)\]
\[4)\ \lim\ _{x\ \to\ a}\ [\frac{f(x)}{g(x)}]\ =\ \frac{\lim\ _{x\ \to\ a}\ f(x)}{\ lm\ _{x\ \to\ a}\ g(x)}\ provided\ \ lm\ _{x\ \to\ a}\ g(x)\ \neq\ 0.\]
\[\color {royalblue} {Formulae}:\ \hspace{20cm}\]
\[1.\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\]
\[2.\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ nθ}{θ}\ =\ n\]
\[\color {royalblue} {Examples}:\ \hspace{20cm}\]
\[\color {purple} {Example\ 1:}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ a}\ \frac{x^3\ -\ a^3}{x\ -\ a}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}\]
\[\lim\ _{x\ \to\ a}\ \frac{x^3\ -\ a^3}{x\ -\ a}\ =\ 3\ a^{3\ -\ 1}\ =\ 3\ a^2\ \hspace{10cm}\]
\[\color {purple} {Example\ 2:}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 2}\ \frac{x^4\ -\ 2^4}{x\ -\ 2}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}\]
\[\lim\ _{x\ \to\ 2}\ \frac{x^4\ -\ a^4}{x\ -\ 2}\ =\ 4\ 2^{4\ -\ 1}\ =\ 4\ 2^3\ =\ 4(8)\ =\ 32\ \hspace{10cm}\]
\[\color {purple} {Example\ 3:}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 2}\ \frac{x^5\ -\ 2^5}{x^3\ -\ 2^3}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}\]
\[Multiply\ and\ divide\ by\ ‘x – 2\]
\[\lim\ _{x\ \to\ 2}\ \frac{\frac{x^5\ -\ 2^5}{x\ -\ 2}}{\frac{x^3\ -\ 2^3}{x\ -\ 2}}\ \hspace{10cm}\]
\[=\ \frac{5\ 2^{5\ -\ 1}}{3\ 2^{3\ -\ 1}}\ \hspace{10cm}\]
\[=\ \frac{5\ 2^4}{3\ 2^2}\ \hspace{10cm}\]
\[=\ \frac{5\ 2^{4\ -\ 2}}{3}\ \hspace{10cm}\]
\[=\ \frac{5\ (2^2)}{3}\ \hspace{10cm}\]
\[=\ \frac{5\ (4)}{3}\ \hspace{10cm}\]
\[=\ \frac{20}{3}\ \hspace{10cm}\]
\[\color {purple} {Example\ 4:}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 3}\ \frac{x^3\ -\ 3^3}{x^4\ -\ 3^4}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}\]
\[Multiply\ and\ divide\ by\ ‘x – 3’\]
\[=\ \lim\ _{x\ \to\ 3}\ \frac{\frac{x^3\ -\ 3^3}{x\ -\ 3}}{\frac{x^4\ -\ 3^4}{x\ -\ 3}}\ \hspace{10cm}\]
\[=\ \frac{3\ (3^{3\ -\ 1})}{4\ (3^{4\ -\ 1})}\ \hspace{10cm}\]
\[=\ \frac{3\ 3^2}{4\ 3^3}\ \hspace{10cm}\]
\[=\ \frac{3}{4}\ 3^{2\ -\ 3}\ \hspace{10cm}\]
\[=\ \frac{3}{4}\ 3^{-1}\ \hspace{10cm}\]
\[=\ \frac{1}{4}\ \hspace{10cm}\]
\[\color {purple} {Example\ 5:}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 3}\ \frac{x^5\ -\ 243}{x^2\ -\ 9}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{x\ \to\ a}\ \frac{x^n\ -\ a^n}{x\ -\ a}\ =\ n\ a^{n\ -\ 1}\ \hspace{15cm}\]
\[\lim\ _{x\ \to\ 3}\ \frac{x^5\ -\ 243}{x^2\ -\ 9}\ =\ \lim\ _{x\ \to\ 3}\ \frac{x^5\ -\ 3^5}{x^2\ -\ 3^2} \hspace{10cm}\]
\[Multiply\ and\ divide\ by\ ‘x – 3’\]
\[=\ \lim\ _{x\ \to\ 3}\ \frac{\frac{x^5\ -\ 3^5}{x\ -\ 3}}{\frac{x^2\ -\ 3^2}{x\ -\ 3}}\ \hspace{10cm}\]
\[=\ \frac{5\ 3^{5\ -\ 1}}{2\ 3^{2\ -\ 1}}\ \hspace{10cm}\]
\[=\ \frac{5\ 3^4}{2\ (3)}\ \hspace{10cm}\]
\[=\ \frac{5\ (3^3)}{2}\ \hspace{10cm}\]
\[=\ \frac{5\ (27)}{2}\ \hspace{10cm}\]
\[=\ \frac{135}{2}\ \hspace{10cm}\]
\[\color {purple} {Example\ 6:}\ \color {Red} {Evaluate:}\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 3θ}{θ}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ nθ}{θ}\ =\ n\ \hspace{15cm}\]
\[\lim\ _{θ\ \to\ 0}\ \frac{Sin\ 3θ}{θ}\ =\ 3\]
\[\color {purple} {Example\ 7:}\ \color {Red} {Evaluate:}\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 10x}{Sin\ 7x}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ nθ}{θ}\ =\ n\ \hspace{15cm}\]
\[Multiply\ and\ divide\ by\ ‘x ‘\]
\[\lim\ _{x\ \to\ 0}\ \frac{\frac{Sin\ 10x}{x}}{\frac{Sin\ 7x}{x}}\ \hspace{10cm}\]
\[=\ \frac{10}{7}\ \hspace{10cm}\]
\[\color {purple} {Example\ 8:}\ \color {Red} {Evaluate:}\ \lim\ _{θ\ \to\ 0}\ \frac{5Sin\ 6θ}{3Sin\ 2θ}\ \hspace{15cm}\]
\[\color {blue}{Solution:}\ W.\ K.\ T\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ nθ}{θ}\ =\ n\ \hspace{15cm}\]
\[Multiply\ and\ divide\ by\ ‘θ ‘\]
\[\lim\ _{θ\ \to\ 0}\ \frac{\frac{5Sin\ 6θ}{θ}}{\frac{3Sin\ 2θ}{θ}}\ \hspace{10cm}\]
\[=\ \frac{5}{3}(\frac{6}{2})\ \hspace{10cm}\]
\[=\ 5\ \hspace{10cm}\]
Exercise Problems
\[\LARGE{\color {purple} {PART- A}}\]
\[\color {purple} {1.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 2}\ \frac{x^2\ -\ 2^2}{x\ -\ 2}\ \hspace{15cm}\]
\[\LARGE{\color {purple} {PART- B}}\]
\[\color {purple} {2.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ a}\ \frac{\sqrt{x}\ -\ \sqrt{a}}{x\ -\ a}\ \hspace{15cm}\]
\[\color {purple} {3.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 7x}{Sin\ 3x}\ \hspace{15cm}\]
\[\color {purple} {4.}\ \color {red} {Evaluate:}\ \lim\ _{x\ \to\ 0}\ \frac{Sin\ 2x}{Sin\ 4x}\ \hspace{15cm}\]
\[\LARGE{\color {purple} {PART- C}}\]
\[\color {purple} {5.}\ \color {red} {Evaluate:}\ (i)\ \lim\ _{x\ \to\ 0}\ \frac{x^2\ +\ 7\ x}{x^2\ +\ 5\ x}\ \hspace{1cm}\ (ii)\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 3\ θ}{Sin\ 2\ θ}\ \hspace{7cm}\]
\[\color {purple} {6.}\ \color {red} {Evaluate:}\ (i)\ \lim\ _{x\ \to\ 3}\ \frac{x^6\ -\ 3^6}{x\ -\ 3}\ \hspace{1cm}\ (ii)\ \lim\ _{θ\ \to\ 0}\ \frac{Sin\ 7\ θ}{Sin\ 2\ θ}\ \hspace{7cm}\]
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