Respuesta :

VectorFundament120

Hi! Your answer is 3x²

Please read an explanation for a clear understanding to the problem.

Any questions about the answer/explanation can be asked through comments! :)

Step-by-step explanation:

Goal

  • Find the limit of f(x+h)-f(x)/h when h --> 0

Given

  • A Cubic Function

[tex]\LARGE{f(x)=x^{3}-9}[/tex]

Step 1

  • Rewrite the limit

[tex]\LARGE{\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}}\\[/tex]

Since f(x) = x³-9. Therefore, f(x+h) would be (x+h)³-9

[tex]\LARGE{\lim_{h \to 0} \frac{[(x+h)^3-9]-(x^3-9)}{h}}[/tex]

Simplify the numerator

[tex]\LARGE{\lim_{h \to 0} \frac{x^3+3x^2h+3xh^2+h^3-9-x^3+9}{h}}\\\LARGE{\lim_{h \to 0} \frac{3x^2h+3xh^2+h^3}{h}}[/tex]

Step 2

  • Factor the numerator so we don't let h = 0.

When finding a limit of function, we can't let the approaching variable equal to 0 (Unless if a function doesn't really have limits.)

[tex]\LARGE{\lim_{h \to 0} \frac{3x^2h+3xh^2+h^3}{h}}\\\LARGE{\lim_{h \to 0} \frac{h(3x^2+3xh+h^2)}{h}}[/tex]

Cancel both h-term from denominator and numerator

[tex]\LARGE{\lim_{h \to 0} \frac{h(3x^2+3xh+h^2)}{h}}\\\LARGE{\lim_{h \to 0} \frac{1(3x^2+3xh+h^2)}{1}}\\\LARGE{\lim_{h \to 0} (3x^2+3xh+h^2)[/tex]

Step 3

  • Substitute h = 0 in the expression.

[tex]\LARGE{\lim_{h \to 0} (3x^2+3xh+h^2)}\\\LARGE{\lim_{h \to 0} (3x^2+3x(0)+0^2)}\\\LARGE{\lim_{h \to 0} (3x^2+0+0)}\\\LARGE{\lim_{h \to 0} 3x^2[/tex]

Since we can't proceed anymore, therefore. The answer is 3x²

Additional Information

  • This is the limit method to find a derivative of function.
  • To find a derivative for polynomial without using limit method, we can do by let exponent become a coefficient then subtract exponent by 1. For example, if you want to differentiate x³ the answer will be 3x².