How can I obtain Leontief and Cobb-Douglas production function from CES function? In most Microeconomics textbooks it is mentioned that the Constant Elasticity of Substitution (CES) production function,
$$Q=\gamma[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{1}{\rho}}$$

(where the elasticity of substitution is $\sigma = \frac 1{1+\rho},\rho &gt; -1$), has as its limits both the Leontief production function and the Cobb-Douglas one. Specifically,

$$\lim_{\rho\to \infty}Q= \gamma \min \left \{K , L\right\}$$

and

$$\lim_{\rho\to 0}Q= \gamma K^aL^{1-a}$$

But they never provide the mathematical proof for these results.  

Can somebody please provide these proofs?  

Moreover, the above CES function incorporates constant-returns-to-scale (homogeneity of degree one), due to the outside exponent being $-1/\rho$. If it was, say $-k/\rho$, then the degree of homogeneity would be $k$.  

How are the limiting results affected if $k\neq 1$?
The proofs I will present are based on techniques relevant to the fact that the CES production function has the form of a generalized weighted mean.
This was used in the original paper where the CES function was introduced, Arrow, K. J., Chenery, H. B., Minhas, B. S., &amp; Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The Review of Economics and Statistics, 225-250.
The authors there referred their readers to the book Hardy, G. H., Littlewood, J. E., &amp; Pólya, G. (1952). Inequalities , chapter $2 $.

We consider the general case
$$Q_k=\gamma[a K^{-\rho} +(1-a) L^{-\rho}  ]^{-\frac{k}{\rho}},\;\; k&gt;0$$

$$\Rightarrow \gamma^{-1}Q_k = \frac 1{[a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{k}{\rho}}}$$

1) Limit when $\rho \rightarrow \infty$
Since we are interested in the limit when $\rho\rightarrow \infty$ we can ignore the interval for which $\rho \leq0$, and treat $\rho$ as strictly positive.

Without loss of generality, assume $K\geq L \Rightarrow (1/K^{\rho})\leq (1/L^{\rho})$. We also have $K, L &gt;0$. Then we verify that the following inequality holds:

$$(1-a)^{k/\rho}(1/L^{k})\leq  \gamma Q_k^{-1} \leq (1/L^{k}) $$

$$\implies (1-a)^{k/\rho}(1/L^{k})\leq  [a (1/K^{\rho}) +(1-a) (1/L^{\rho})  ]^{\frac{k}{\rho}} \leq (1/L^{k}) \tag{1}$$

by raising throughout to the $\rho/k$ power to get

$$(1-a)(1/L^{\rho}) \leq a (1/K^{\rho}) +(1-a) (1/L^{\rho})  \leq (1/L^{\rho}) \tag {2}$$
which indeed holds, obviously, given the assumptions. Then go back to the first element of $(1)$ and

$$\lim_{\rho\rightarrow \infty} (1-a)^{k/\rho}(1/L^{k}) =(1/L^{k})$$

which sandwiches the middle term in $(1)$ to $(1/L^{k})$ , so 

$$\lim_{\rho\rightarrow \infty}Q_k = \frac {\gamma }{1/L^k} = \gamma L^k = {\gamma }\big[\min\{K,L\}\big]^{k} \tag{3}$$

So for $k=1$ we obtain the basic Leontief production function.  

2) Limit when $\rho \rightarrow 0$
Write the function using exponential as

$$\gamma^{-1}Q_k=\exp\left\{-\frac k{\rho}\cdot \ln\big[a (K^{\rho})^{-1} +(1-a) (L^{\rho})^{-1}\big]\right\} \tag {4}$$

Consider the first-order Maclaurin expansion (Taylor expansion centered at zero) of the term inside the logarithm, with respect to $\rho$:

$$a (K^{\rho})^{-1} +(1-a) (L^{\rho})^{-1} \\= a (K^{0})^{-1} +(1-a) (L^{0})^{-1} -a (K^{0})^{-2}K^{0}\rho\ln K- (1-a) (L^{0})^{-2}L^{0}\rho\ln L + O(\rho^2) \\$$

$$=1 - \rho a\ln K - \rho(1-a)\ln L+ O(\rho^2) = 1 +\rho \big[\ln K^{-a}L^{-(1-a)}\big]+ O(\rho^2)$$ 

Insert this back into $(4)$ and get rid of the outer exponential,

$$\gamma^{-1}Q_k = \left(1 +\rho \big[\ln K^{-a}L^{-(1-a)}\big]+ O(\rho^{2})\right)^{-k/\rho}$$

In case it is opaque, define $r\equiv 1/\rho$ and re-write

$$\gamma^{-1}Q_k = \left(1 +\frac{\big[\ln K^{-a}L^{-(1-a)}\big]}{r}+ O(r^{-2})\right)^{-kr}$$

Now it does look like an expression whose limit at infinity will give us something exponential:

$$\lim_{\rho\rightarrow 0}\gamma^{-1}Q_k = \lim_{r\rightarrow \infty}\gamma^{-1}Q_k = \left(\exp\left\{ \ln K^{-a}L^{-(1-a)}\right\} \right)^{-k}$$

$$\Rightarrow \lim_{\rho\rightarrow 0}Q_k =\gamma\left(K^{a}L^{1-a}\right)^k$$

The degree of homogeneity $k$ of the function is preserved, and if $k=1$ we obtain the Cobb-Douglas function.  

It was this last result that made Arrow and Co to call $a$ the "distribution" parameter of the CES function.