Question

$$\lim_{h \to 0} \frac{5(x + h)^3 - 5x^3}{h}$$

Explanation:

Step1: Expand \((x + h)^3\)

$$(x + h)^3=x^3+3x^2h + 3xh^2+h^3$$
So, \(5(x + h)^3-5x^3=5(x^3+3x^2h + 3xh^2+h^3)-5x^3\)
$$=5x^3+15x^2h + 15xh^2+5h^3-5x^3=15x^2h + 15xh^2+5h^3$$

Step2: Simplify the fraction

\(\frac{5(x + h)^3-5x^3}{h}=\frac{15x^2h + 15xh^2+5h^3}{h}\)
Since \(h
eq0\) (as \(h
ightarrow0\) but \(h
eq0\) in the limit process), we can factor out \(h\) from the numerator:
\(\frac{h(15x^2 + 15xh+5h^2)}{h}=15x^2 + 15xh+5h^2\)

Step3: Evaluate the limit

\(\lim_{h
ightarrow0}(15x^2 + 15xh+5h^2)\)
Substitute \(h = 0\) into the expression \(15x^2 + 15xh+5h^2\)
When \(h = 0\), we get \(15x^2+15x\times0 + 5\times0^2=15x^2\)

Answer:

\(15x^2\)