The value of the following limit is _____________.

$$\mathop {\lim }\limits_{x \to {0^ + }} {{\sqrt x } \over {1 - {e^{2\sqrt x }}}}$$

Which one of the following is the closed form for the generating function of the sequence (a_n}_{n $$\ge$$ 0} defined below?

$${a_n} = \left\{ {\matrix{ {n + 1,} & {n\,is\,odd} \cr {1,} & {otherwise} \cr } } \right.$$

Consider a simple undirected unweighted graph with at least three vertices. If A is the adjacency matrix of the graph, then the number of 3-cycles in the graph is given by the trace of

Consider solving the following system of simultaneous equations using LU decomposition.

x₁ + x₂ $$-$$ 2x₃ = 4

x₁ + 3x₂ $$-$$ x₃ = 7

2x₁ + x₂ $$-$$ 5x₃ = 7

where L and U are denoted as

$$L = \left( {\matrix{ {{L_{11}}} & 0 & 0 \cr {{L_{21}}} & {{L_{22}}} & 0 \cr {{L_{31}}} & {{L_{32}}} & {{L_{33}}} \cr } } \right),\,U = \left( {\matrix{ {{U_{11}}} & {{U_{12}}} & {{U_{13}}} \cr 0 & {{U_{22}}} & {{U_{23}}} \cr 0 & 0 & {{U_{33}}} \cr } } \right)$$

Which one of the following is the correct combination of values for L₃₂, U₃₃, and x₁ ?