Question

❖ᴏɴʟʏ ᴘʀᴏᴘᴇʀ ꜱᴏʟᴠᴇᴅ ᴀɴꜱᴡᴇʀ ᴡɪᴛʜ ɢᴏᴏᴅ ᴇxᴘʟᴀɴᴀɪᴏɴ ɴᴇᴇᴅᴇᴅ
❖ ɴᴏ ꜱᴘᴀᴍᴍɪɴɢ
❖ᴏɴʟʏ ꜰᴏʀ ᴍᴏᴅᴇʀᴀᴛᴏʀꜱ, ʙʀᴀɪɴʟʏ ꜱᴛᴀʀꜱ ᴀɴᴅ ᴏᴛʜᴇʀ ʙᴇꜱᴛ ᴜꜱᴇʀꜱ​

Answer 1

Method -1:

${\boxed{\boxed{\begin{array}{cc}\bf \: \to \: \: given : \\ \\ \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \sqrt{x + 6} - 2 }{x + 2} \right) \\ \\ \small{ \sf \: if \: we \: apply \: limit \: we \: will \: get \: \frac{0}{0} \: \: form} \\ \sf \: which \: is \: undefined.\\ \\ \bf \: so \: simplify \: the \: limit \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{( \sqrt{x + 6} - 2)( \sqrt{x + 6} + 2)}{(x + 2)( \sqrt{x + 6} + 2) } \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{( { \sqrt{x + 6} })^{2} - {2}^{2} }{(x + 2)( \sqrt{x + 6} + 2)} \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{x + 6 - 4}{(x + 2)( \sqrt{x + 6} + 2) } \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \cancel{(x + 2)}}{ \cancel{(x + 2)}( \sqrt{x + 6} + 2) } \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{1}{ \sqrt{x + 6} + 2} \right) \\ \\ \bf \: apply \: limit \\ \\ = \frac{1}{ \sqrt{ - 2 + 6} + 2 } \\ \\ = \frac{1}{ \sqrt{4} + 2} \\ \\ = \frac{1}{2 + 2} \\ \\ = \frac{1}{4} \\ \\ \\ \blue{ \boxed{ \therefore \: \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \sqrt{x + 6} - 2}{x + 2} \right) = \frac{1}{4} }}\end{array}}}}$

Method -2: La'Hospital rule

Rule: L'Hospital's Rule tells us that if we have an indeterminate form 0/0 or ∞/∞ all we need to do is differentiate the numerator and differentiate the denominator and then take the limit.

$\orange{\boxed{\boxed{\begin{array}{cc}\bf \: \to \:given : \\ \\ \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \sqrt{x + 6} - 2 }{x + 2} \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{\frac{d}{dx} ( \sqrt{x + 6} - 2)}{ \frac{d}{dx}(x + 2) }\right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \frac{d}{dx} \sqrt{x + 6} - \frac{d}{dx} 2}{ \frac{d}{dx}x + \frac{d}{dx}2 } \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \frac{1}{2 \sqrt{x + 6}}. \frac{d}{dx}(x + 6) - 0}{1 + 0} \right) \\ \\ = \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{1}{2 \sqrt{x + 6} } \times 1\right) \\ \\ = \frac{1}{2 \sqrt{ - 2 + 6} } \\ \\ = \frac{1}{2 \sqrt{4} } \\ \\ = \frac{1}{2 \times 2} \\ \\ = \frac{1}{4} \\ \\ \\ \blue{ \boxed{ \therefore \: \displaystyle \lim_{ \rm \: x \to \: - 2} \: \rm \: \left( \frac{ \sqrt{x + 6} - 2}{x + 2} \right) = \frac{1}{4} }} \end{array}}}}$