qu.1.topic=1-experiments@ qu.1.1.question=Compute ${mathml((1-cos ($m*x))/x^2)} for ${mathml(x)} values of 0.1, 0.01, and 0.001.
Use these values to guess the exact value of

\lim_{x \to 0} \frac{1 - \cos ($m x)}{x^{2}}

.@ qu.1.1.answer.num=$answer@ qu.1.1.answer.units=@ qu.1.1.showUnits=false@ qu.1.1.grading=exact_value@ qu.1.1.negStyle=both@ qu.1.1.numStyle= @ qu.1.1.mode=Numeric@ qu.1.1.comment=The values of the function are getting closer and closer to a simple number.@ qu.1.1.editing=useHTML@ qu.1.1.algorithm=$m=range(2,9); $answer=0.5*$m^2;@ qu.1.2.question=Estimate

\lim_{x \to 0} \frac{{$n}^{x} - 1}{x}

. Report your answer to the nearest 0.001.@ qu.1.2.answer.num=$answer@ qu.1.2.answer.units=@ qu.1.2.showUnits=false@ qu.1.2.grading=toler_abs@ qu.1.2.err=0.001@ qu.1.2.negStyle=both@ qu.1.2.numStyle= @ qu.1.2.mode=Numeric@ qu.1.2.comment=The answer is $answer.@ qu.1.2.editing=useHTML@ qu.1.2.algorithm=$n=range(2,8); $answer=ln($n);@ qu.2.topic=2-computing limits by factoring@ qu.2.1.mode=Multipart@ qu.2.1.editing=useHTML@ qu.2.1.algorithm=$a=range(1,9); $answer=$a;@ qu.2.1.question=@ qu.2.1.weighting=1,1@ qu.2.1.numbering=alpha@ qu.2.1.part.1.editing=useHTML@ qu.2.1.part.1.choice.4=-$a@ qu.2.1.part.1.question=Let

f (x)=

${mathml((x^2-$a*x)/(x-$a))}. What is

f ($a)

?@ qu.2.1.part.1.choice.3=undefined@ qu.2.1.part.1.choice.2=0@ qu.2.1.part.1.choice.1=$a@ qu.2.1.part.1.comment=What happens when you substitute $a into ${mathml(f)}(${mathml(x)})?@ qu.2.1.part.1.mode=Multiple Choice@ qu.2.1.part.1.algorithm=@ qu.2.1.part.1.answer=3@ qu.2.1.part.2.editing=useHTML@ qu.2.1.part.2.question=Compute

\lim_{x \to $a}

${mathml((x^2-$a*x)/(x-$a))}.@ qu.2.1.part.2.algorithm=@ qu.2.1.part.2.answer=$answer@ qu.2.1.part.2.mode=Formula@ qu.2.1.part.2.comment=Remove the common factor of ${mathml(x-$a)}. You can do this because ${mathml(x)} is not equal to $a.@ qu.2.2.mode=Formula@ qu.2.2.comment=The answer is ${mathml("($b-$a)"/"($c-$a)")}.@ qu.2.2.editing=useHTML@ qu.2.2.algorithm=$a=range(-5,5); $b=range(-5,5); $c=range(-5,5); condition:not(eq($a*$b*$c*($b-$a)*($c-$a)*($c-$b),0)); $answer="($b-$a)"/"($c-$a)";@ qu.2.2.question=Compute

\lim_{x \to $a}

${mathml((x^2-($a+$b)*x+$a*$b)/(x^2-($a+$c)*x+$a*$c))}.@ qu.2.2.answer=$answer@ qu.3.topic=3-continuity@ qu.3.1.mode=Multipart@ qu.3.1.editing=useHTML@ qu.3.1.algorithm=$a=range(-1,1,2); $b=range(1,4); $m=switch(rint(4),-3,-2,2,3); $n=switch(rint(4),-3,-2,2,3); $c=$m*$a+$b-$n*$a; condition:not(eq($m,$n));condition:gt($c,0); $answer=$m*$a+$b;@ qu.3.1.question=Let

f (x) = {\begin{matrix} $m x + $b & x < $a \\ $n x + $c & x > $a \end{matrix}

.@ qu.3.1.weighting=1,1,2@ qu.3.1.numbering=alpha@ qu.3.1.part.1.editing=useHTML@ qu.3.1.part.1.question=Compute

\lim_{x \to {$a}^{-}} f (x)

.@ qu.3.1.part.1.algorithm=@ qu.3.1.part.1.answer=$answer@ qu.3.1.part.1.mode=Formula@ qu.3.1.part.1.comment=The answer is $answer.@ qu.3.1.part.2.editing=useHTML@ qu.3.1.part.2.question=Compute

\lim_{x \to {$a}^{+}} f (x)

.@ qu.3.1.part.2.algorithm=@ qu.3.1.part.2.answer=$answer@ qu.3.1.part.2.mode=Formula@ qu.3.1.part.2.comment=The answer is $answer.@ qu.3.1.part.3.editing=useHTML@ qu.3.1.part.3.choice.4=There is a non-removable discontinuity at

$a

.@ qu.3.1.part.3.question=Which of the following statements are true?@ qu.3.1.part.3.choice.3=There is a removable discontinuity at

$a

.@ qu.3.1.part.3.choice.2=${mathml(f)} is continuous at

$a

.@ qu.3.1.part.3.choice.1=${mathml(f)} is continuous at all points

x \neq $a

.@ qu.3.1.part.3.comment=

f ($a)

is not defined.@ qu.3.1.part.3.mode=Multiple Selection@ qu.3.1.part.3.answer=1, 3@ qu.3.2.mode=Multipart@ qu.3.2.editing=useHTML@ qu.3.2.algorithm=$a=range(-1,1,2); $b=range(1,4); $c=range(1,4); $d=range(1,4); $m=switch(rint(4),-3,-2,2,3); $n=switch(rint(4),-3,-2,2,3); $p=switch(rint(4),-3,-2,2,3); condition:not(eq($n*$a+$c,($m*$a+$b)*($p*$a-$d))); condition:not(eq($d,$p*$a)); $answera=$m*$a+$b; $answerb="($n*$a+$c)/($p*$a-$d)";@ qu.3.2.question=Let

f (x) = {\begin{matrix} $m x + $b & x \leq $a \\ \frac{$n x + $c}{$p x - $d} & x > $a \end{matrix}

.@ qu.3.2.weighting=1,1,2@ qu.3.2.numbering=alpha@ qu.3.2.part.1.editing=useHTML@ qu.3.2.part.1.question=Compute

\lim_{x \to {$a}^{-}} f (x)

.@ qu.3.2.part.1.algorithm=@ qu.3.2.part.1.answer=$answera@ qu.3.2.part.1.mode=Formula@ qu.3.2.part.1.comment=The answer is $answera.@ qu.3.2.part.2.editing=useHTML@ qu.3.2.part.2.question=Compute

\lim_{x \to {$a}^{+}} f (x)

.@ qu.3.2.part.2.algorithm=@ qu.3.2.part.2.answer=$answerb@ qu.3.2.part.2.mode=Formula@ qu.3.2.part.2.comment=The answer is ${mathml("($n*$a+$c)/($p*$a-$d)")}.@ qu.3.2.part.3.editing=useHTML@ qu.3.2.part.3.choice.4=There is a non-removable discontinuity at

$a

.@ qu.3.2.part.3.question=Which of the following statements are true?@ qu.3.2.part.3.choice.3=There is a removable discontinuity at

$a

.@ qu.3.2.part.3.choice.2=${mathml(f)} is continuous at

$a

.@ qu.3.2.part.3.choice.1=${mathml(f)} is continuous at all points

x \neq $a

.@ qu.3.2.part.3.comment=The limits as

x \to $a

from the left and right do not agree.@ qu.3.2.part.3.mode=Multiple Selection@ qu.3.2.part.3.answer=4@ qu.3.3.mode=Multipart@ qu.3.3.editing=useHTML@ qu.3.3.algorithm=$a=range(-1,1,2); $b=range(1,4); $d=range(1,4); $m=switch(rint(4),-3,-2,2,3); $n=switch(rint(4),-3,-2,2,3); $p=switch(rint(4),-3,-2,2,3); $c=($m*$a+$b)*($p*$a-$d)-$n*$a; condition:gt($c,0); condition:gt($a,$d/$p); $answer=$m*$a+$b;@ qu.3.3.question=Let

f (x) = {\begin{matrix} $m x + $b & x \leq $a \\ \frac{$n x + $c}{$p x - $d} & x > $a \end{matrix}

.@ qu.3.3.weighting=1,1,2@ qu.3.3.numbering=alpha@ qu.3.3.part.1.editing=useHTML@ qu.3.3.part.1.question=Compute

\lim_{x \to {$a}^{-}} f (x)

.@ qu.3.3.part.1.algorithm=@ qu.3.3.part.1.answer=$answer@ qu.3.3.part.1.mode=Formula@ qu.3.3.part.1.comment=The answer is $answer.@ qu.3.3.part.2.editing=useHTML@ qu.3.3.part.2.question=Compute

\lim_{x \to {$a}^{+}} f (x)

.@ qu.3.3.part.2.algorithm=@ qu.3.3.part.2.answer=$answer@ qu.3.3.part.2.mode=Formula@ qu.3.3.part.2.comment=The answer is $answer.@ qu.3.3.part.3.editing=useHTML@ qu.3.3.part.3.choice.4=There is a non-removable discontinuity at

$a

.@ qu.3.3.part.3.question=Which of the following statements are true?@ qu.3.3.part.3.choice.3=There is a removable discontinuity at

$a

.@ qu.3.3.part.3.choice.2=${mathml(f)} is continuous at

$a

.@ qu.3.3.part.3.choice.1=${mathml(f)} is continuous at all points

x \neq $a

.@ qu.3.3.part.3.comment=The formula for

x > $a

has a discontinuity, but the discontinuity is at a point where the formula is not used to define

f (x)

.@ qu.3.3.part.3.mode=Multiple Selection@ qu.3.3.part.3.answer=1, 2@ qu.3.4.mode=Multipart@ qu.3.4.editing=useHTML@ qu.3.4.algorithm=$a=range(-1,1,2); $b=range(1,4); $d=range(1,4); $m=switch(rint(4),-3,-2,2,3); $n=switch(rint(4),-3,-2,2,3); $p=switch(rint(4),-3,-2,2,3); $c=($m*$a+$b)*($p*$a-$d)-$n*$a; condition:gt($c,0); condition:lt($a,$d/$p); $answer=$m*$a+$b;@ qu.3.4.question=Let

f (x) = {\begin{matrix} $m x + $b & x \leq $a \\ \frac{$n x + $c}{$p x - $d} & x > $a \end{matrix}

.@ qu.3.4.weighting=1,1,2@ qu.3.4.numbering=alpha@ qu.3.4.part.1.editing=useHTML@ qu.3.4.part.1.question=Compute

\lim_{x \to {$a}^{-}} f (x)

.@ qu.3.4.part.1.algorithm=@ qu.3.4.part.1.answer=$answer@ qu.3.4.part.1.mode=Formula@ qu.3.4.part.1.comment=The answer is $answer.@ qu.3.4.part.2.editing=useHTML@ qu.3.4.part.2.question=Compute

\lim_{x \to {$a}^{+}} f (x)

.@ qu.3.4.part.2.algorithm=@ qu.3.4.part.2.answer=$answer@ qu.3.4.part.2.mode=Formula@ qu.3.4.part.2.comment=The answer is $answer.@ qu.3.4.part.3.editing=useHTML@ qu.3.4.part.3.choice.4=There is a non-removable discontinuity at

$a

.@ qu.3.4.part.3.question=Which of the following statements are true?@ qu.3.4.part.3.choice.3=There is a removable discontinuity at

$a

.@ qu.3.4.part.3.choice.2=${mathml(f)} is continuous at

$a

.@ qu.3.4.part.3.choice.1=${mathml(f)} is continuous at all points

x \neq $a

.@ qu.3.4.part.3.comment=Make sure you look for all possible discontinuities.@ qu.3.4.part.3.mode=Multiple Selection@ qu.3.4.part.3.answer=2@ qu.4.topic=4-limits at infinity@ qu.4.1.mode=Formula@ qu.4.1.comment=The answer is ${mathml("$a/$c")}.@ qu.4.1.editing=useHTML@ qu.4.1.algorithm=$p=range(2,5); $a=range(1,5); $b=range(-5,5); $c=range(1,5); $d=range(-5,5); condition:not(eq($d,0)); $answer="$a/$c";@ qu.4.1.question=Find

\lim_{x \to ∞}

${mathml(($a*x^$p+$b)/($c*x^$p+$d))}.@ qu.4.1.answer=$answer@ qu.4.2.mode=Formula@ qu.4.2.comment=The answer is $answer.@ qu.4.2.editing=useHTML@ qu.4.2.algorithm=$p=range(2,4); $q=$p+1; $a=range(1,5); $b=range(-5,5); $c=range(1,5); $d=range(-5,5); condition:not(eq($d,0)); $answer=0;@ qu.4.2.question=Find

\lim_{x \to ∞}

${mathml(($a*x^$p+$b)/($c*x^$q+$d))}.@ qu.4.2.answer=$answer@ qu.4.3.mode=Formula@ qu.4.3.comment=The answer is ${mathml("$a/$c")}.@ qu.4.3.editing=useHTML@ qu.4.3.algorithm=$a=range(-5,5); $c=range(1,5); $d=range(-5,5); condition:not(eq($a*$d,0)); $answer="$a/$c";@ qu.4.3.question=Find

\lim_{x \to ∞}

${mathml(($a*x)/sqrt($c^2*x^2+$d))}.@ qu.4.3.answer=$answer@