How Does the Gamma Function Simplify to 'z' in the Limit?

[Muthumanimaran]

This is given in Mathematical methods for physicists by Arfken and weber, while defining a property of gamma function, I have no idea how the term in the red circle becomes 'z' in the final step, please help

 

[gleem]

Try multiplying the numerator and denominator by z and pull out an n from the exponent then rearrange terms.

 

[Muthumanimaran]

I'm asking about the final step, How to get the final step from the second step?

 

[gleem]

The limit of the product is product of the limit of the factors. What is the definition of the gamma function?

 

[Muthumanimaran]

I know the limit of the second term in the product converges to "gamma of z', it is how the gamma function is defined, But I couldn't figure out how the limit of the first term in the product converges to 'z'. How do I compute it?

 

[HallsofIvy]

So you are asking specifically about
\lim_{n\to \infty} \frac{nz}{z+ n+ 1}

Divide both numerator and denominator by n:
\lim_{n\to\infty}\frac{z}{\frac{z}{n}+ 1+ \frac{1}{n}}

 

[gleem]

nz/(n+z+1) → nz/n in the limit n→∞

 

[Muthumanimaran]

So you are asking specifically about
\lim_{n\to \infty} \frac{nz}{z+ n+ 1}

Divide both numerator and denominator by n:
\lim_{n\to\infty}\frac{z}{\frac{z}{n}+ 1+ \frac{1}{n}}

Yeah that's what I asked for, Thank you!