limx→−∞√x2+x5+3x

The given expression is: lim (x→−∞) √(x² + x⁵ + 3x)

First, we can factor out x² from the square root to simplify the expression:

lim (x→−∞) √(x²(1 + x³ + 3/x))

This simplifies to:

lim (x→−∞) |x|√(1 + x³ + 3/x)

Since x is approaching negative infinity, |x| will be -x. So, the expression becomes:

lim (x→−∞) -x√(1 + x³ + 3/x)

As x approaches negative infinity, the term 3/x will approach 0. So, the expression simplifies to:

lim (x→−∞) -x√(1 + x³)

As x approaches negative infinity, the term 1 will become insignificant compared to x³. So, the expression simplifies to:

lim (x→−∞) -x√(x³)

This simplifies to:

lim (x→−∞) -x²√x

As x approaches negative infinity, √x will approach infinity. So, the final answer is:

-∞