displaystyle limx → 2 ((ex-e2/x-2))=

Question

displaystyle limx → 2 ((ex-e2/x-2))= (A) 0 (B) 1 (C) e2 (D) e

Tardigrade Admin · Accepted Answer

Put x - 2 = y, so that when x → 2 then y → 0. ∴ displaystyle limx → 2 ((ex-e2/x-2)) = displaystyle limy → 0((ey+2-e2/y)) = displaystyle limy → 0 e2⋅((ey-1/y)) =e2⋅ displaystyle limy → 0((ey-1/y)) =e2×1=e2

Q. $x \to 2 lim$ $(\frac{e ^{x} - e ^{2}}{x - 2}) =$

$0$

$1$

$e^{2}$

$e$

Put $x - 2 = y$ , so that when $x \to 2$ then $y \to 0$ .
$∴ x \to 2 lim$ $(\frac{e ^{x} - e ^{2}}{x - 2})$
$= y \to 0 lim$ $(\frac{e ^{y + 2} - e ^{2}}{y})$
$= y \to 0 lim$ ${e^{2} \cdot (\frac{e ^{y} - 1}{y})}$
$= e^{2} \cdot y \to 0 lim$ $(\frac{e ^{y} - 1}{y})$
$= e^{2} \times 1 = e^{2}$

Q. x→2lim​ (x−2ex−e2​)=

0

1

e2

e

Put x−2=y, so that when x→2 then y→0. ∴x→2lim​ (x−2ex−e2​) =y→0lim​(yey+2−e2​) =y→0lim​{e2⋅(yey−1​)} =e2⋅y→0lim​(yey−1​) =e2×1=e2

Q. $x \to 2 lim$ $(\frac{e ^{x} - e ^{2}}{x - 2}) =$

$0$

$1$

$e^{2}$

$e$

Put $x - 2 = y$ , so that when $x \to 2$ then $y \to 0$ .
$∴ x \to 2 lim$ $(\frac{e ^{x} - e ^{2}}{x - 2})$
$= y \to 0 lim$ $(\frac{e ^{y + 2} - e ^{2}}{y})$
$= y \to 0 lim$ ${e^{2} \cdot (\frac{e ^{y} - 1}{y})}$
$= e^{2} \cdot y \to 0 lim$ $(\frac{e ^{y} - 1}{y})$
$= e^{2} \times 1 = e^{2}$