Question

Prove :- $\bf \displaystyle \lim_{x\to a} \bf \bigg( \dfrac{x²-a²}{x-a} \bigg) = 2a [tex]\ \textless \ br /\ \textgreater \ \ \textless \ br /\ \textgreater \ From epsilon delta definition of limit which states that , for any [tex] \varepsilon > 0$ , there exist a +ve no. $\delta$ such that ${ \bf 0 < | x - a | < \delta }$ , then ${ \bf | f ( x ) - l | < \varepsilon }$

Range of x and f(x) are as follows

$\leadsto \bf f ( x ) \in ( l - \varepsilon , l + \varepsilon )$

$\leadsto \bf x \in ( a - \delta , a + \delta )$ ​

Answer 1

very easy, bro

Step-by-step explanation:

$\bf \displaystyle \lim_{x\to a} \bf \bigg( \dfrac{x²-a²}{x-a} \bigg) \\ = \bf \displaystyle \lim_{x\to a} \bf \bigg( \dfrac{(x + a)(x - a)}{x-a} \bigg) \\ =\bf \displaystyle \lim_{x\to a} \bf \bigg(x + a \bf \bigg) \\ = a + a = 2a$

Answer 2

Hello ItzHeartRider!

Here's your solution~

$\boxed{\mathfrak{{Answer \: with \: explanation}}} \downarrow$

$\sf {\displaystyle \lim _{ \sf \:x \to a}{\bigg(}\sf{\frac{x {}^{2} - {a}^{2} }{x - a} }\bigg)}$

$\sf \longmapsto \displaystyle \lim _{\sf {x \to a}} \bigg( \sf{\frac{(x + a)(x - a)}{x - a} \bigg)}$

$\sf \longmapsto \displaystyle \lim _{\sf {x \to a}}\bigg(\sf {x + a}\bigg)$

$\sf \longmapsto \: a + a = 2a$

Therefore, the answer is:

$\sf \boxed{{\sf\red{ a + a = 2a}}}$

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More to know:

Trigonometry Table:

$\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\mathfrak\red{ Trigonometry\: Table} \\ \begin{gathered}\begin{gathered}\begin{gathered}\begin{gathered}\boxed{\boxed{\begin{array}{ |c |c|c|c|c|c|} \bf\angle A & \bf{0}^{ \circ} & \bf{30}^{ \circ} & \bf{45}^{ \circ} & \bf{60}^{ \circ} & \bf{90}^{ \circ} \\ \\ \rm sin A & 0 & \dfrac{1}{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{ \sqrt{3}}{2} &1 \\ \\ \rm cos \: A & 1 & \dfrac{ \sqrt{3} }{2}& \dfrac{1}{ \sqrt{2} } & \dfrac{1}{2} &0 \\ \\ \rm tan A & 0 & \dfrac{1}{ \sqrt{3} }&1 & \sqrt{3} & \rm \infty \\ \\ \rm cosec A & \rm \infty & 2& \sqrt{2} & \dfrac{2}{ \sqrt{3} } &1 \\ \\ \rm sec A & 1 & \dfrac{2}{ \sqrt{3} }& \sqrt{2} & 2 & \rm \infty \\ \\ \rm cot A & \rm \infty & \sqrt{3} & 1 & \dfrac{1}{ \sqrt{3} } & 0 \end{array}}}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}\end{gathered}\end{gathered}\end{gathered} \end{gathered}\end{gathered}$

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Extra:

What is Trigonometry?

• Trigonometry is a field of mathematics.
• There are Six functions of Trigonometry.
• It has formulas that we have to learn.
• It relates between the angles and sides (Of any triangle)

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