1. **Stating the problem:** Evaluate the limit $$\lim_{x\to \infty} \left(1 + \frac{5}{3x}\right)^{\frac{x}{2}}.$$\n\n2. **Rewrite the expression:** Notice the limit has the form similar to $$\lim_{n\to \infty} \left(1 + \frac{a}{n}\right)^n = e^a.$$ Here, the base is $$1 + \frac{5}{3x}$$ and the exponent is $$\frac{x}{2}$$ instead of $$x$$.\n\n3. **Manipulate the exponent:** Rewrite the expression so it resembles the standard exponential limit.\n\n$$ \left(1 + \frac{5}{3x}\right)^{\frac{x}{2}} = \left[\left(1 + \frac{5}{3x}\right)^{3x/5}\right]^{\frac{5}{6}} $$\nHere, we multiplied and divided the exponent by $$\frac{3}{5}$$: $$\frac{x}{2} = \frac{5}{6} \times \frac{3x}{5}.$$\n\n4. **Evaluate the inner limit:** \n$$ \lim_{x \to \infty} \left(1 + \frac{5}{3x}\right)^{3x/5} = e,$$ because this is of the form $$\lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n = e$$ with $$n = \frac{3x}{5}$$.\n\n5. **Calculate the overall limit:** \nSince the inner expression tends to $$e$$, we raise it to the power $$\frac{5}{6}$$: \n$$ \lim_{x \to \infty} \left(1 + \frac{5}{3x}\right)^{\frac{x}{2}} = e^{\frac{5}{6}}.$$\n\n**Final answer:** \n$$ \boxed{e^{\frac{5}{6}}}. $$