limit + integral

**Columbo** · October 22nd 2008, 05:52 PM

Hey,
Any help would be appreciated.
I'm not even sure where to start.

Compute

lim (x^2)/x-3 [Integral] (from 3 to x) (sin t)/t dt
x->3

**skeeter** · October 22nd 2008, 06:02 PM

L'Hopital?

$\lim_{x \to 3} \frac{x^2 \cdot \frac{\sin{x}}{x} + 2x \cdot \int_3^x \frac{\sin{t}}{t}\,dt}{1}$

**Krizalid** · October 22nd 2008, 06:52 PM

Power series allowed?

I haven't checked it well though.

**o_O** · October 22nd 2008, 06:58 PM

Let $F(x) = \int_{3}^{x} \frac{\sin t}{t} \ dt$ .

Note: $F(3) = 0$

So: $\lim_{x \to 3} \frac{x^2}{x-3} \cdot \int_{3}^{x} \frac{\sin t}{t} \ dt \ = \ \lim_{x \to 3} x^2 \cdot \frac{F(x) - F(3)}{x - 3} \ = \ \lim_{x \to 3} x^2 \ \cdot \ \underbrace{\lim_{x \to 3} \frac{F(x) - F(3)}{x - 3}}_{F'(3)} \ = \ \hdots$

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