1. HaǐTH lim _(xarrow -infty )(1+(2)/(x))^x+5

The given problem is to find the limit as approaches negative infinity of the expression \((1+\frac{2}{x})^x + 5\). This is a classic limit problem involving an exponential expression.To solve this, we need to recognize that as approaches negative infinity, approaches 0. Therefore, the base of the exponential expression approaches 1.The expression \((1 + \frac{2}{x})^x\) can be compared to the well-known limit \(\lim_{x \to \infty}(1 + \frac{1}{x})^x\), which is the definition of , the base of natural logarithms. However, in our case, since approaches negative infinity, the expression \((1 + \frac{2}{x})^x\) approaches because the exponent is negative, which flips the base of the exponential expression.Finally, we add 5 to this limit, as indicated in the original expression, to get the final result.