lim-x-0-0-x-2-4-t-3-dt-x-2-

Question Number 78526 by TawaTawa last updated on 18/Jan/20

\lim_{x \to 0} [\frac{\int_{0}^{\boldsymbol x^{2}} \sqrt{4 + \boldsymbol t^{3}} \boldsymbol dt}{\boldsymbol x^{2}}]

Commented by mr W last updated on 18/Jan/20

\lim_{x \to 0} [\frac{\int_{0}^{\boldsymbol x^{2}} \sqrt{4 + \boldsymbol t^{3}} \boldsymbol dt}{\boldsymbol x^{2}}]

= \lim_{x \to 0} [\frac{\sqrt{4 + {(x^{2})}^{3}} 2 x}{2 x}]

= \lim_{x \to 0} [\sqrt{4 + x^{6}}]

= \sqrt{4}

= 2

Commented by TawaTawa last updated on 18/Jan/20

God bless you sir .

Commented by mr W last updated on 18/Jan/20

w h a t a b o u t

\lim_{x \to 0} [\frac{\int_{x}^{\boldsymbol x^{2}} \sqrt{4 + \boldsymbol t^{3}} \boldsymbol dt}{\boldsymbol x^{2}}] ?

Commented by TawaTawa last updated on 18/Jan/20

I was about to ask a question sir .

did you use the integration sign . how come you get \sqrt{4 + {(x^{2})}^{3}} \times 2 x

Commented by TawaTawa last updated on 18/Jan/20

I will solve your new question sir, if i understand how

you integrate

Commented by mr W last updated on 18/Jan/20

You can't use 'macro parameter character #' in math mode

Commented by john santu last updated on 18/Jan/20

\lim_{x \to 0} \frac{\sqrt{4 + x^{6}} \times 2 x - \sqrt{4 + x^{3}} \times 1}{2 x}

= \lim_{x \to 0} \sqrt{4 + x^{6}} - \lim_{x \to 0} \frac{x \sqrt{\frac{4}{x^{2}} + x}}{2 x} = - \infty

Commented by mr W last updated on 18/Jan/20

c o r r e c t s i r!

b u t b o t h - \infty a n d + \infty, s o j u s t = \infty .

Commented by jagoll last updated on 18/Jan/20

y e s s i r

Commented by TawaTawa last updated on 18/Jan/20

Checked sir, i get the approach now .

= \lim_{x \to 0} [\frac{\sqrt{4 + {(x^{2})}^{3}} \times 2 x - \sqrt{4 + x^{3}} \times 1}{2 x}] {The power of x decreases}

= [\frac{\sqrt{4 + 0} \times 2 (0) - \sqrt{4 + 0}}{2 (0)}]

= [\frac{- \sqrt{4}}{2 (0)}]

= - \frac{2}{0}

= \infty

Commented by mathmax by abdo last updated on 18/Jan/20

l e t f (x) = \frac{1}{x^{2}} \int_{x}^{x^{2}} \sqrt{4 + t^{3}} d t

\exists c_{x} \in] x, x^{2} [/ f (x) = \frac{\sqrt{4 + c_{x}^{2}}}{x^{2}} \int_{x}^{x^{2}} d t = \sqrt{4 + c_{x}^{2}} \times \frac{x^{2} - x}{x^{2}}

= \sqrt{4 + c_{x}^{2}} \times (1 - \frac{1}{x}) \Rightarrow {l i m}_{x \to 0^{+}} f (x) = - \infty

Commented by TawaTawa last updated on 18/Jan/20

God bless you sir .

God bless you sir .

Commented by john santu last updated on 19/Jan/20

t h a n k s y o u

t h a n k s y o u

Commented by mathmax by abdo last updated on 19/Jan/20

y o u a r e w e l c o m e

y o u a r e w e l c o m e