Question 1001202
The square of the quantity 2 plus h plus 3 times the quantity 2 plus h plus 1
is {{{(2+h)^2+3*(2+h)+1=f(2+h)}}} .
The quantity 2 squared plus 3 times 2 plus 1
is {{{2^2+3*2+1=f(2)=11}}} .



CHOICE # 1:
The limit as h approaches 0 of
the quotient of
the square of the quantity 2 plus h plus 3 times the quantity 2 plus h plus 1 
minus
the quantity 2 squared plus 3 times 2 plus 1,
and h
is {{{lim(h->0,(((2+h)^2+3*(2+h)+1)-(2^2+3*2+1))/h)=lim(h->0,(f(2+h)-f(2))/h)=lim(h->0,(f(2+h)-11)/h)}}} .
That is by definition {{{"f'(2)"}}} .


CHOICE # 2 does not find the value of {{{"f'(2)"}}} :
The limit as h approaches 0 of
the quotient of
the square of the quantity 2 plus h plus 3 times the quantity 2 plus h plus 1 
minus 2 squared
plus 3 times 2
plus 1,
and h
is {{{lim(h->0,(((2+h)^2+3*(2+h)+1)-2^2+3*2+1)/h)=lim(h->0,(f(2+h)-4+6+1)/h)=lim(h->0,(f(2+h)+3)/h)}}} .
That is not {{{"f'(2)"=lim(h->0,(f(2+h)-11)/h)}}} .


CHOICE # 3:
The limit as x approaches 2 of
the quotient of
x squared plus 3 times x plus 1
minus 11,
and
the quantity x minus 2
is {{{lim(x->2,((x^2+3x+1)-11)/(x-2))}}}={{{lim(x->2,(f(x)-11)/(x-2))}}}={{{lim(x->2,(f(x)-f(2))/(x-2))}}} .
That is by definition {{{"f'(2)"}}} .