lim-x-1-4x-lnx-x-1-

Question Number 97577 by Rio Michael last updated on 08/Jun/20

\lim_{x \to 1} \frac{4 x \ln x}{x - 1} = ? ?

Commented by Dwaipayan Shikari last updated on 23/Jun/20

4 \lim_{x \to 1} \frac{x l o g (1 + x - 1)}{x - 1} = 4

Answered by smridha last updated on 08/Jun/20

\frac{0}{0} f o r m u s i n g L^{'} H \hat{o p i t a l} r u l e

4 \lim_{x \to 1} \frac{[1 + l n (x)]}{1} = 4

Commented by Rio Michael last updated on 08/Jun/20

thanks! please can you use another method

other thank L^{'} {Hopital}^{'} s rule

Commented by smridha last updated on 08/Jun/20

o k k

\lim_{x \to 1} \frac{4 x l n x}{x - 1} = \lim_{h \to 0} \frac{4 (h + 1) l n (h + 1)}{h}

= 4 \lim_{h \to 0} l n (h + 1) + 4 \lim_{h \to 0} \frac{l n (1 + h)}{h}

= 0 + 4 \times 1 = 4

Commented by Rio Michael last updated on 08/Jun/20

thank you

Answered by abdomathmax last updated on 08/Jun/20

changement x - 1 = t give \frac{4 xlnx}{x - 1} = \frac{4 (t + 1) \ln (1 + t)}{t}

t \to 1 \Rightarrow t \to 0 so \frac{4 (t + 1) \ln (1 + t)}{t} \sim \frac{4 (t + 1) t}{t}

= 4 (t + 1) \to 4 \Rightarrow \lim_{x \to 1} \frac{4 xlnx}{x - 1} = 4