Question: \[\lim_{x\to\infty}\frac{e^{6x}-e^{-6x}}{e^{6x}+e^{-6x}}\]
1. **State the problem:** We want to find the limit \(\lim_{x\to\infty}\frac{e^{6x}-e^{-6x}}{e^{6x}+e^{-6x}}\).
2. **Recall the behavior of exponential functions:** As \(x \to \infty\), \(e^{6x} \to \infty\) and \(e^{-6x} \to 0\).
3. **Rewrite the expression by factoring \(e^{6x}\) from numerator and denominator:**
$$\frac{e^{6x}-e^{-6x}}{e^{6x}+e^{-6x}} = \frac{e^{6x}(1 - e^{-12x})}{e^{6x}(1 + e^{-12x})}$$
4. **Cancel \(e^{6x}\) from numerator and denominator:**
$$\frac{\cancel{e^{6x}}(1 - e^{-12x})}{\cancel{e^{6x}}(1 + e^{-12x})} = \frac{1 - e^{-12x}}{1 + e^{-12x}}$$
5. **Evaluate the limit as \(x \to \infty\):** Since \(e^{-12x} \to 0\), the expression becomes
$$\frac{1 - 0}{1 + 0} = 1$$
6. **Final answer:**
$$\lim_{x\to\infty}\frac{e^{6x}-e^{-6x}}{e^{6x}+e^{-6x}} = 1$$