Answer:
[tex]4x[/tex]
Step-by-step explanation:
Step 1: Define
I will substitute delta x with h
[tex]\lim_{h \to 0} \frac{2(x+h)^2-2x^2}{h}[/tex]
Step 2: Simplify function
- Expand: [tex]\lim_{h \to 0} \frac{2(x^2+2hx+h^2)-2x^2}{h}[/tex]
- Distribute 2: [tex]\lim_{h \to 0} \frac{2x^2+4hx+2h^2-2x^2}{h}[/tex]
- Combine like terms: [tex]\lim_{h \to 0} \frac{4hx+2h^2}{h}[/tex]
- Factor: [tex]\lim_{h \to 0} \frac{h(4x+2h)}{h}[/tex]
- Simplify: [tex]\lim_{h \to 0} 4x+2h[/tex]
Step 3: Evaluate limit
[tex]\lim_{h \to 0} 4x+2h=\lim_{h \to 0} 4x+2(0)[/tex]
[tex]\lim_{h \to 0} 4x+2(0)=\lim_{h \to 0} 4x+0[/tex]
[tex]\lim_{h \to 0} 4x+0 = 4x[/tex]
