Question

Find the limit (if it exists). $$ \lim _{\Delta x \rightarrow 0} \frac{(x+\Delta x)^{3}-x^{3}}{\Delta x} $$

Short answer

The answer is \(3x^2\).

Step-by-step solution

Express as a Difference of Cubes

Express \( (x+\Delta x)^{3}-x^{3} \) as \( (x+\Delta x - x)((x+\Delta x)^{2} + (x+\Delta x)x + x^{2}) \). This is the formula for the difference of cubes.

Simplify the Equation

Simplify the above equation as \( \Delta x((x+\Delta x)^{2} + (x+\Delta x)x + x^{2}) \). Here \(\Delta x\) is common in both numerator and denominator, so it can be cancelled out giving us a resulting equation of \((x+\Delta x)^{2} + (x+\Delta x)x + x^{2} \).

Apply the Limit

Now we apply \( \lim _{\Delta x \rightarrow 0} \) to the equation \((x+\Delta x)^{2} + (x+\Delta x)x + x^{2} \). As \( \Delta x \rightarrow 0 \), we replace \( \Delta x \) with 0 to get the result.