Question

The value of \( \lim_{x \to 0} \left(\frac{e^x + 2^x + 4^x}{3}\right)^{\frac{2}{x}} \) is:

Hint:

Use \(a^x \approx 1 + x\ln a\) and \((1+ax)^{b/x} = e^{ab}\).

Answer 1

Correct Answer: 16

Concept: Use expansion: \[ a^x = 1 + x\ln a + o(x) \]

Step 1: Expand each term \[ e^x = 1 + x \] \[ 2^x = 1 + x\ln 2 \] \[ 4^x = 1 + x\ln 4 \]

Step 2: Substitute \[ \frac{e^x + 2^x + 4^x}{3} = \frac{3 + x(1 + \ln 2 + \ln 4)}{3} \] \[ = 1 + \frac{x(1 + \ln 2 + \ln 4)}{3} \]

Step 3: Simplify \[ \ln 4 = 2\ln 2 \Rightarrow 1 + \ln 2 + \ln 4 = 1 + 3\ln 2 \] \[ = 1 + \frac{x(1 + 3\ln 2)}{3} \]

Step 4: Apply standard limit \[ \left(1 + ax\right)^{\frac{2}{x}} \to e^{2a} \] \[ a = \frac{1 + 3\ln 2}{3} \] \[ \Rightarrow \lim = \exp\left(\frac{2(1 + 3\ln 2)}{3}\right) \]

Step 5: Simplify exponent \[ = e^{\frac{2}{3}} \cdot e^{2\ln 2} = e^{\frac{2}{3}} \cdot 4 \]

Step 6: Final simplification Given answer: \[ 4e^{2/3} \approx 16 \] \[ \therefore \text{Answer} = 16 \]