Limits and Derivatives: common misconceptions

If your child says… “Any joining line gives the exact speed at the first point.”

Ask together

Does the slope change when you move the second point closer?

What’s true

A joining line gives an interval’s average rate.

If your child says… “The value at zero is one, so the limit must be one.”

Ask together

Do the arrows follow the diagonal towards the hole or towards the filled point?

What’s true

A value assigned at one input does not control nearby outputs.

If your child says… “I can divide the limits even when the bottom approaches zero.”

Ask together

Would dividing by zero produce a number?

What’s true

The division rule requires a nonzero limit underneath.

If your child says… “Zero over zero means the limit is zero.”

Ask together

After cancellation, what does the unchanged nearby expression approach?

What’s true

Zero over zero alone tells us no limit value.

If your child says… “Any two bounds tell us the middle limit.”

Ask together

If the bounds stay apart, what forces the middle towards just one number?

What’s true

The bounds must approach the same number to force one middle limit.

If your child says… “I should put the input change equal to zero immediately.”

Ask together

What happens to the first denominator if you put zero in immediately?

What’s true

Keep the change nonzero during division, then take its limit.

If your child says… “I multiply the two derivatives to differentiate a product.”

Ask together

Where does the changing size of each factor appear if you multiply only their derivatives?

What’s true

Each changing factor contributes a term while the other factor remains undifferentiated.

If your child says… “Cosine differentiates to positive sine.”

Ask together

Would positive sine match the falling cosine tangent at half pi radians?

What’s true

Cosine’s derivative is minus sine when the angle is measured in radians.