BeMath-1-lim-b-a-b-a-a-b-a-a-b-a-2a-a-2-lim-x-0-3e-2x-e-x-4-x-

Question Number 108623 by bemath last updated on 18/Aug/20

\frac{\supset B e M a t h \supset}{★}

(1) \lim_{b \to a} \frac{b \sqrt{a} - a \sqrt{b}}{a \sqrt{a} + b \sqrt{a} - 2 a \sqrt{a}}

(2) \lim_{x \to 0} \frac{3 e^{2 x} + e^{x} - 4}{x}

Answered by ajfour last updated on 18/Aug/20

\lim \frac{b \sqrt{a} - a \sqrt{b}}{a \sqrt{a} + b \sqrt{a} - 2 a \sqrt{a}}

L = \lim_{b \to a} (\frac{{a b}^{2} - a^{2} b}{a^{3} + {a b}^{2} + 2 a^{2} b - 4 a^{3}}) \times \lim_{b \to a} (\frac{a \sqrt{a} + b \sqrt{a} + 2 \sqrt{a}}{b \sqrt{a} + a \sqrt{b}})

= \lim_{b \to a} \frac{b (b - a)}{b^{2} + 2 a b - 3 a^{2}} \times 2

= \lim_{b \to a} \frac{b (b - a)}{(b + 3 a) (b - a)} \times 2

\Rightarrow L = \frac{1}{2}

Commented by bemath last updated on 18/Aug/20

c o o l l

Answered by ajfour last updated on 18/Aug/20

L = \lim_{x \to 0} \frac{3 e^{2 x} + e^{x} - 4}{x}

L = \lim_{x \to 0} \frac{3 (e^{2 x} - 1) + (e^{x} - 1)}{x}

L = 3 \times 2 + 1 = 7 .

Answered by mathmax by abdo last updated on 18/Aug/20

2) f (x) = \frac{{3 e}^{2 x} + e^{x} - 4}{x} we hsve e^{2 x} \sim 1 + 2 x and e^{x} \sim 1 + x \Rightarrow

f (x) \sim \frac{3 + 6 x + 1 + x - 4}{x} = 7 (x \to 0) \Rightarrow \lim_{x \to 0} f (x) = 7