**Analysis of the Function \( f(x) = x^2 + x^{-1} + e^{2x} + e^{-2x} \)**The following discusses the behavior of each term in the function \( f(x) \) as \( x \) approaches infinity (\( \infty \)) and negative infinity (\( -\infty \)).### Behavior as \( x \to \infty \):1. **\( x^2 \)** - As \( x \to \infty \), \( x^2 \to \infty \).2. **\( x^{-1} \)** - As \( x \to \infty \), \( x^{-1} \to 0 \).3. **\( e^{2x} \)** - As \( x \to \infty \), \( e^{2x} \to \infty \).4. **\( e^{-2x} \)** - As \( x \to \infty \), \( e^{-2x} \to 0 \).### Behavior as \( x \to -\infty \):1. **\( x^2 \)** - As \( x \to -\infty \), \( x^2 \to \infty \).2. **\( x^{-1} \)** - As \( x \to -\infty \), \( x^{-1} \to 0 \).3. **\( e^{2x} \)** - As \( x \to -\infty \), \( e^{2x} \to 0 \).4. **\( e^{-2x} \)** - As \( x \to -\infty \), \( e^{-2x} \to \infty \).### Leading Behavior of \( f(x) \) - **At \( \infty \):** - Dominated by \( e^{2x} \to \infty \).- **At \( -\infty \):** - Dominated by \( e^{-2x} \to \infty \). These observations help in understanding the growth rate and dominant behaviors of different terms as \( x \) approaches infinity or negative infinity.

as x tends to infinity , x^2 tends to infinity As x tends to infinity , x^-(1) =1/x tends to zero As x tends to infinity ,e^2x tends to infinity As x tends to infinity , e^(-2x) tends to zeroAs x tends to - infinity ,x^2 tends to infinity As x tends to - infinity , x^-(1) =1/x tends to zero As x tends to - infinity ,e^2x tends to zero As x tends to - infinity , e^(-2x) tends to infinity Lim x tends to infinity f(x)=infinity Lim x tends to - infinity f(x)= infinity