Answer:
C. 2916
Step-by-step explanation:
The given limits is
[tex]\lim_{h \to 0} \frac{f(9+h)-f(9)}{h}[/tex]
if [tex]f(x)=x^4[/tex].
[tex]\Rightarrow f(9)=9^4=6561[/tex]
[tex]f(h+9)=(h+9)^4=h^4+36 h^3+486 h^2+2916 h+6561[/tex]
Our limit becomes;
[tex]\lim_{h \to 0} \frac{f(h+9)-f(9)}{h}= \lim_{h \to 0} \frac{h^4+36 h^3+486 h^2+2916 h+6561-6561}{h}[/tex]
This simplifies to;
[tex]\lim_{h \to 0} \frac{f(h+9)-f(9)}{h}= \lim_{h \to 0} \frac{h^4+36 h^3+486 h^2+2916 h}{h}[/tex]
[tex]\lim_{h \to 0} \frac{f(h+9)-f(9)}{h}= \lim_{h \to 0} h^3+36 h^2+486 h+2916 [/tex]
[tex]\lim_{h \to 0} \frac{f(h+9)-f(9)}{h}= (0)^3+36 (0)^2+486(0)+2916 [/tex]
[tex]\lim_{h \to 0} \frac{f(h+9)-f(9)}{h}= 2916 [/tex]
the correct choice is C.
