I am confused about direct sub. How did you get that.[math]\lim_{x \to 1}\exp\left\{\cot(\pi x)\log(\sin(\pi x)+1)\right\}=\\ \lim_{x \to 1}\exp\left\{\frac{\cos(\pi x)}{\sin(\pi x)}\log(\sin(\pi x)+1)\right\}=\\ \underbrace{\lim_{x \to 1}\exp\{\cos(\pi x)\}}_{\text{direct sub}}\cdot \underbrace{ \lim_{x \to 1}\exp\left\{\frac{\log(\sin(\pi x)+1)}{\sin(\pi x)}\right\}}_{\text{L'Hopital}}[/math]
The notes are for you what to do, not how I got there.I am confused about direct sub. How did you get that
I don't understand how you got the third row.Let [imath]\displaystyle y = \lim_{x \to 1} [1+\sin(\pi x)]^{\cot(\pi x)}[/imath]
[imath]\displaystyle \ln{y} = \lim_{x \to 1} \ln\bigg([1+\sin(\pi x)]^{\cot(\pi x)} \bigg)[/imath]
[imath]\displaystyle \ln{y} = \lim_{x \to 1} \dfrac{\ln[1+\sin(\pi x)]}{\tan(\pi x)}[/imath]
now try applying L'Hopital
I don't understand them.Its
The notes are for you what to do, not how I got there.
[math]\lim g(x)h(x)=\lim g(x)\times \lim h(x)[/math]I don't understand them.
What do I do with that?[math]\lim g(x)h(x)=\lim g(x)\times \lim h(x)[/math]
You asked my how I got the direct substitution part in post #2. That’s the property I applied. Look at post 2 again.What do I do with that?
I don't get it. Could you write it a bit more detailed.You asked my how I got the direct substitution part in post #2. That’s the property I applied. Look at post 2 again.
[imath]\displaystyle \ln{y} = \lim_{x \to 1} \ln[1 + \sin(\pi x)]^{\cot(\pi x)}[/imath]I don't understand how you got the third row.
How did you get lny?[imath]\displaystyle \ln{y} = \lim_{x \to 1} \ln[1 + \sin(\pi x)]^{\cot(\pi x)}[/imath]
[imath]\displaystyle \ln{y} = \lim_{x \to 1} \cot(\pi x) \cdot \ln[1 + \sin(\pi x)][/imath]
[imath]\displaystyle \ln{y} = \lim_{x \to 1} \dfrac{1}{\tan(\pi x)} \cdot \ln[1 + \sin(\pi x)][/imath]
...
Look at response #3How did you get lny?
Can you just "multiply" both sides with ln even with limits involved?Look at response #3
it’s not multiplying … you’re taking the natural log of both sidesCan you just "multiply" both sides with ln even with limits involved?
What does natural log mean?it’s not multiplying … you’re taking the natural log of both sides
And why does it work while my way does notWhat does natural log mean?
What you’re doing is not different than what I’m doing. You need to do an extra step of algebraic manipulation.And why does it work while my way does not