Calculating Limits and Integrals: Tips and Tricks for Solving Tough Problems

[TheMadHathor]

"Some really hard problems" That is exactly what our teacher wrote on the top of this worksheet we got for homework. I'm in Calc AB and our teacher gave us 12 really evil problems. I'm stuck just on the first problem:

Evaluate lim (x/x-3)(Intergral of sint/t dt from 3->x)
x->3

Sorry if the notation is kind of weird, I don't know how to get half these symbols on the computer.

Anyway, I first tried to find the limit of just x/x-3 and it's undefined. Then I tried finding the intergral of sint/t dt but it just kept going around in a circle from (ln t)(sin t)-the intergral of ln t (-cos t)-(1/t)(-cos t)-the intergral of 1/t sin t...and so forth. I believe i saw somewhere that if an intergral keeps repeating it's undefined, but my teacher hinted that there's a definite solution to it.

Can someone help me? This assignment is due Thursday, and I haven't a clue as how to go about solving these. I'll need help on some of the other problems too, which I'll post later. Someone please respond soon! I don't want to fail!

 

[dextercioby]

Can someone help me? This assignment is due Thursday, and I haven't a clue as how to go about solving these. I'll need help on some of the other problems too, which I'll post later. Someone please respond soon! I don't want to fail!

I'm trying to help uopefully i' right.I'm not a mathematician.
$$\lim_{x\rightarrow 3}\frac{x}{x-3}\int_{3}^{x} \frac{\sin t}{t} dt$$=...??( 1)

Define this function (called "sine integral")
$$Si(x)=:\int_{0}^{x} \frac{\sin t}{t} dt$$
Using the additivity property of the definite integral,it's not hard to show that the integral in (1) is
$$Si(x)-Si(3)$$

Your limit becomes
$$\lim_{x\rightarrow 3} x\frac{Si(x)-Si(3)}{x-3}$$
Now use the definition for the derivative of a function in one point to get the limit
$$3 \frac{d Si(x)}{dx}|_{x=3}$$.

Use the fact that:
$$\lim_{t\rightarrow 0}\frac{\sin t}{t} =1$$ and the fact that:
$$\frac{d}{dx}[\int_{a}^{x} f(t) dt]=f(x)$$

To get the answer:
$$3\frac{\sin 3}{3}=\sin 3$$

However,i'm not sure on this result.Mathematicians on this forum can correct my mistakes and if,by chance,everything i did is correct,supply you with rigurous mathematical founding to my calculations.

Daniel.

 

[Tide]

... or, you could just apply l'Hopital directly to your expression - integral and all! :-)

 

[dextercioby]

... or, you could just apply l'Hopital directly to your expression - integral and all! :-)

I don't know,Tide,my approach seems elegant. No L'Hospital,no tricky differentiations,just definitions and a simple limit.

Daniel.

 

[Tide]

I don't know,Tide,my approach seems elegant. No L'Hospital,no tricky differentiations,just definitions and a simple limit.

Daniel.

A thousand pardons! I failed to recognize a masterpiece when I saw it!

 

[Hurkyl]

Anyways, the lesson to be learned is that, no matter how complicated it looks,

$$\int_a^x f(t) \, dt$$

is merely a function of x.

 

[TheMadHathor]

Thanks! L'Hopital's theorem makes it so much easier, duh! It was the next section in our book!

 

[dextercioby]

Thanks! L'Hopital's theorem makes it so much easier, duh! It was the next section in our book!

So much for the elegance in mathematics... Anyway,u're free to post your problems.

Daniel.