1. **Problem statement:** We want to find a number $\delta$ such that if $|x - 4| < \delta$, then $|\sqrt{x} - 2| < 0.4$. This means we want to control how close $\sqrt{x}$ is to 2 when $x$ is close to 4. 2. **Recall the function and values:** The function is $f(x) = \sqrt{x}$ and at $x=4$, $f(4) = \sqrt{4} = 2$. 3. **Rewrite the inequality:** We want $$|\sqrt{x} - 2| < 0.4.$$ This means $$1.6 < \sqrt{x} < 2.4.$$ 4. **Square all parts to remove the square root:** Since the square root function is increasing for $x \geq 0$, squaring preserves inequalities: $$1.6^2 < x < 2.4^2,$$ which gives $$2.56 < x < 5.76.$$ 5. **Express this interval in terms of $|x - 4|$:** We want $x$ to be within some distance $\delta$ from 4, so $$|x - 4| < \delta.$$ From the interval above, the closest endpoints to 4 are 2.56 and 5.76. Calculate distances: $$4 - 2.56 = 1.44,$$ $$5.76 - 4 = 1.76.$$ 6. **Choose the smaller distance for $\delta$:** To ensure $x$ stays in the interval, $$\delta = \min(1.44, 1.76) = 1.44.$$ 7. **Conclusion:** If $|x - 4| < 1.44$, then $|\sqrt{x} - 2| < 0.4$. **Final answer:** $$\boxed{\delta = 1.44}.$$