Question

lim x-> 2 [( x³ + √( x+2) - 10/ x² - 4]

solve it​

Answer 1

Answer:

$\large\boxed{\sf{\dfrac{49}{16}}}$

Explanation:

Given a limit such that,

$\displaystyle \lim_{x\to2} \dfrac{ {x}^{3} + \sqrt{x + 2} - 10 }{ {x}^{2} - 4 }$

To solve the above limit,

Let's apply L Hospitals rule.

We have to differentiate both numerator and denominator separately.

Therefore, we will get,

$= \displaystyle \lim_{x\to2} \frac{ \frac{d}{dx} {x}^{3} + \frac{d}{dx} \sqrt{x + 2} - \frac{d}{dx} 10}{ \frac{d}{dx} {x}^{2} - \frac{d}{dx}4 }$

But, we know that,

• $\dfrac{d}{dx} {x}^{n} = n {x}^{n - 1}$

• $\dfrac{d}{dx} c = 0$

Where, c is any constant term.

Therefore, we will get,

$= \displaystyle \lim_{x\to2} \dfrac{3 {x}^{2} + \frac{1}{2 \sqrt{x + 2} } }{2x}$

Now, substituting the limits, we get,

$= \dfrac{3 \times {2}^{2} + \frac{1}{2 \sqrt{2 + 2} } }{2 \times 2} \\ \\ = \frac{ 12 + \frac{1}{4}}{4} \\ \\ = \frac{48 + 1}{4 \times 4} \\ \\ = \frac{49}{16}$

Hence, the required value is $\bold{\dfrac{49}{16}}$

Answer 2

Answer is given in the attachment.