1. **Problem Statement:** Find the limit of a function using a change of variable. 2. **General Idea:** When evaluating limits, sometimes substituting a new variable simplifies the expression. For example, if the limit involves $x \to a$, and the expression is complicated, set $t = g(x)$ such that as $x \to a$, $t \to b$. Then rewrite the limit in terms of $t$. 3. **Formula:** If $\lim_{x \to a} f(x)$ is difficult, and $t = g(x)$ with $\lim_{x \to a} g(x) = b$, then $$\lim_{x \to a} f(x) = \lim_{t \to b} f(g^{-1}(t))$$ 4. **Example:** Suppose we want to find $$\lim_{x \to 0} \frac{\sin(3x)}{x}$$ 5. **Change of Variable:** Let $t = 3x$. Then as $x \to 0$, $t \to 0$. 6. **Rewrite Limit:** $$\lim_{x \to 0} \frac{\sin(3x)}{x} = \lim_{t \to 0} \frac{\sin(t)}{t/3} = \lim_{t \to 0} \frac{3\sin(t)}{t}$$ 7. **Evaluate Limit:** Using the standard limit $\lim_{t \to 0} \frac{\sin t}{t} = 1$, we get $$\lim_{t \to 0} \frac{3\sin(t)}{t} = 3 \times 1 = 3$$ 8. **Answer:** The limit is $3$. This method simplifies the evaluation by substituting a variable to use known limits.