Cracking The Ap Calculus ab Exam 2018

SOLUTIONS TO PRACTICE PROBLEM SET 17

1.

Recall  L’Hôpital’s Rule:   If  f(c)    =   g(c)    =   0,  or  if  f(c)    =   g(c)    =   ∞,  and if  f′(c)   and g′(c)   exist,  and

if  g′(c)   ≠   0,  then     .  Here    f(x)    =   sin 3x, and g(x)    =   sin 4x, and sin 0   =   0.  This

means    that    we  can     use     L’Hôpital’s     Rule    to  find    the     limit.  We  take    the     derivative  of  the

numerator   and the denominator:     .  If  we  take    the new limit,  we  get 

= .

2. −1

Recall  L’Hôpital’s Rule:   If  f(c)    =   g(c)    =   0,  or  if  f(c)    =   g(c)    =   ∞,  and if  f′(c)   and g′(c)   exist,  and

if  g′(c)   ≠   0,  then     .  Here    f(x)    =   x   −   π   and g(x)    =   sin x.  We  can see that    x   −   π

=   0   when    x   =   π,  and that    sinπ    =   0.  This    means   that    we  can use L’Hôpital’s Rule    to  find    the limit.

We  take    the derivative  of  the numerator   and the denominator:     .  If  we

take    the new limit,  we  get .

3.

Recall  L’Hôpital’s Rule:   If  f(c)    =   g(c)    =   0,  or  if  f(c)    =   g(c)    =   ∞,  and if  f′(c)   and g′(c)   exist,  and

if  g′(c)   ≠   0,  then     .  Here    f(x)    =   x   −   sin x   and g(x)    =   x^3 .   We  can see that    x   −

sin x   =   0   whenx   =   0,  and that    0^3     =   0.  This    means   that    we  can use L’Hôpital’s Rule    to  find    the

limit.   We  take    the     derivative  of  the     numerator   and     the     denominator:   

. If we take the new limit, we get =

. But this is still indeterminate, so what do we do? Use L’Hôpital’s Rule