1. **State the problem:** Find the limit \(\lim_{x \to 0} \frac{(x+1)^5 - 1}{x}\). 2. **Rewrite the expression:** The numerator is \((x+1)^5 - 1\), and the denominator is \(x\). 3. **Apply the Binomial theorem:** Expand \((x+1)^5\): $$ (x+1)^5 = 1 + 5x + 10x^2 + 10x^3 + 5x^4 + x^5 $$ 4. **Simplify numerator:** Subtract 1 from the expansion: $$ (x+1)^5 - 1 = 5x + 10x^2 + 10x^3 + 5x^4 + x^5 $$ 5. **Form the fraction:** $$ \frac{(x+1)^5 - 1}{x} = \frac{5x + 10x^2 + 10x^3 + 5x^4 + x^5}{x} $$ 6. **Simplify the fraction:** Divide numerator terms by \(x\): $$ 5 + 10x + 10x^2 + 5x^3 + x^4 $$ 7. **Calculate the limit as \(x \to 0\):** Substitute \(x=0\) into simplified expression: $$ 5 + 10(0) + 10(0)^2 + 5(0)^3 + (0)^4 = 5 $$ **Final answer:** $$\lim_{x \to 0} \frac{(x+1)^5 - 1}{x} = 5$$