1. ## Mth101 Q1_8

(a) Find the Limit using L’ Hospital’s rule
lim x→0 (x-tanx)/sinx-x

(b) show that the given integral converges or not
∫+∞e 1/ln3x dx

2. Originally Posted by TAHIR
(a) Find the Limit using L’ Hospital’s rule
lim x→0 (x-tanx)/sinx-x
Let $L = \lim_{x \to 0} \frac {x - \tan x}{ \sin x - x}$ ........This goes to $\frac {0}{0}$

Apply L'Hopital's, we get:

$L = \lim_{x \to 0} \frac {1 - \sec^2 x}{ \cos x - 1}$ .......Again, this goes to $\frac {0}{0}$

Apply L'Hopital's again, we get:

$L = \lim_{x \to 0} \frac {2 \sec^2 x \tan x}{ \sin x}$ .......This again goes to $\frac {0}{0}$, but L'Hopital's would be overkill here, let's just factor

$L = \lim_{x \to 0} \frac {2 \sec^2 x }{ \cos x}$

$\Rightarrow L = 2$