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   $\underset{x\to 0}{\mathrm{lim}}\frac{1-\mathrm{cos}\left(\mathrm{m}x\right)}{{x}^{2}}$.@ 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 $\underset{x\to 0}{\mathrm{lim}}\frac{{\mathrm{n}}^{x}-1}{x}$.   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   $f\left(x\right)=$ \${mathml((x^2-\$a*x)/(x-\$a))}.   What is $f\left(\mathrm{a}\right)$?@ 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   $\underset{x\to a}{\mathrm{lim}}$ \${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   $\underset{x\to a}{\mathrm{lim}}$ \${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 $f\left(x\right)=\left\{\begin{array}{cc}\mathrm{m}x+\mathrm{b}& x<\mathrm{a}\\ \mathrm{n}x+\mathrm{c}& x>\mathrm{a}\end{array}$.@ 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 $\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)$.@ 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 $\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)$.@ 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 $x\ne a$.@ qu.3.1.part.3.comment= $f\left(\mathrm{a}\right)$ 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 $f\left(x\right)=\left\{\begin{array}{cc}\mathrm{m}x+\mathrm{b}& x\le \mathrm{a}\\ \frac{\mathrm{n}x+\mathrm{c}}{\mathrm{p}x-\mathrm{d}}& x>\mathrm{a}\end{array}$.@ 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 $\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)$.@ 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 $\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)$.@ 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 $x\ne a$.@ qu.3.2.part.3.comment=The limits as $x\to \mathrm{a}$ 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 $f\left(x\right)=\left\{\begin{array}{cc}\mathrm{m}x+\mathrm{b}& x\le \mathrm{a}\\ \frac{\mathrm{n}x+\mathrm{c}}{\mathrm{p}x-\mathrm{d}}& x>\mathrm{a}\end{array}$.@ 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 $\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)$.@ 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 $\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)$.@ 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 $x\ne a$.@ qu.3.3.part.3.comment=The formula for $x>\mathrm{a}$ has a discontinuity, but the discontinuity is at a point where the formula is not used to define $f\left(x\right)$.@ 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 $f\left(x\right)=\left\{\begin{array}{cc}\mathrm{m}x+\mathrm{b}& x\le \mathrm{a}\\ \frac{\mathrm{n}x+\mathrm{c}}{\mathrm{p}x-\mathrm{d}}& x>\mathrm{a}\end{array}$.@ 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 $\underset{x\to {a}^{-}}{\mathrm{lim}}f\left(x\right)$.@ 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 $\underset{x\to {a}^{+}}{\mathrm{lim}}f\left(x\right)$.@ 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 $x\ne a$.@ 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 $\underset{x\to \infty }{\mathrm{lim}}$ \${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 $\underset{x\to \infty }{\mathrm{lim}}$ \${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 $\underset{x\to \infty }{\mathrm{lim}}$ \${mathml((\$a*x)/sqrt(\$c^2*x^2+\$d))}.@ qu.4.3.answer=\$answer@