If f x -1- x 1 - x - e - x thenA- lim x - f x - e -2- lim x - 0 fx-e-2- lim x - 0 f x -1D- lim x - 0 f x -1

The correct options are B lim_x → 0 f(x)= e/2 D lim_x → 0 f(x) lt; 1 Limit =lim_x → 0e^1/xln(1+x) e/x =lim_x → 0e^ ( 1 x/2+x^2/3 ) e/x=lim_x → 0e{ e^ ( x/ ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Right Hand Limit

If f x =1+ x ...

Question

If $f (x) = \frac{(1 + x)^{1 / x} - e}{x}$ then

{lim}_{x \to \infty} f (x) = \frac{e}{2}

No worries! We‘ve got your back. Try BYJU‘S free classes today!

{lim}_{x \to 0} f (x) = - \frac{e}{2}

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

{lim}_{x \to 0} f (x) > 1

No worries! We‘ve got your back. Try BYJU‘S free classes today!

{lim}_{x \to 0} f (x) < - 1

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

Open in App

Solution

The correct options are
B ${lim}_{x \to 0} f (x) = - \frac{e}{2}$

D ${lim}_{x \to 0} f (x) < - 1$

Limit = ${lim}_{x \to 0} \frac{e^{\frac{1}{x} l n (1 + x)} - e}{x}$
$= {lim}_{x \to 0} \frac{e^{(1 - \frac{x}{2} + \frac{x^{2}}{3})} - e}{x} = {lim}_{x \to 0} \frac{e ⎧ ⎪ ⎨ ⎪ ⎩ e^{(\frac{- x}{2} + \frac{x^{2}}{3} . . . . .)} - 1 ⎫ ⎪ ⎬ ⎪ ⎭}{(\frac{- x}{2} + \frac{x^{2}}{3} . . . . .)} \times \frac{(\frac{- x}{2} + \frac{x^{2}}{3} . . . .)}{x} = e (l n e) (- \frac{1}{2}) = \frac{- e}{2} < - 1 (∵ e ≅ 2.7)$

Suggest Corrections

Similar questions

If f(x) = $\frac{(1 + x)^{\frac{1}{x}} - e}{x}$ then

Q. If

f (x) = {(\frac{x}{2 + x})}^{2 x},

then

Q. Let

f (x) = g (x) \frac{e^{1 / x} - e^{- 1 / x}}{e^{1 / x} + e^{- 1 / x}}

and

x \neq 0

where g is a continuous function.

Then

{lim}_{x \to 0} f (x)

exists if?

Q. If

f (x) = \{\begin{array}{l} 2 x + 3, & x \leq 0 \\ 3 (x + 1), & x > 0 \end{array} .

Find

\lim_{x \to 0} f (x)

and

\lim_{x \to 1} f (x)

Q. Let

a_{1} > a_{2} > a_{3} > \dots a_{n} > 1;

p_{1} > p_{2} > p_{3} > . . . p_{n} > 0

be such that

p_{1} + p_{2} + p_{3} + \dots + p_{n} = 1.

F (x) = {(p_{1} a_{1}^{x} + p_{2} a_{2}^{x} + \dots + p_{n} a_{n}^{x})}_{1}^{1 / x},

then

Join BYJU'S Learning Program

If f(x)=(1+x)1/x−ex then

The correct options are B limx→0f(x)=−e2 D limx→0f(x)<−1 Limit =limx→0e1xln(1+x)−ex =limx→0e(1−x2+x23)−ex=limx→0e⎧⎪⎨⎪⎩e(−x2+x23.....)−1⎫⎪⎬⎪⎭(−x2+x23.....)×(−x2+x23....)x=e(ln e)(−12)=−e2<−1(∵e≅2.7)

If $f (x) = \frac{(1 + x)^{1 / x} - e}{x}$ then