Limits (10/14/11) Question: How can we compute the

slope of the tangent line to the curve y = x 2 at the point (1, 1)?

Possible approach: Compute the slope of the secant line which the connects the points (1, 1) and (1 + h, (1+ h )2) for small values of h.

Now try to see the limit as h goes toward 0.

Instantaneous Velocity That example was the “Tangent Problem”.

Now comes the “Velocity Problem”. Question: Given the position of a moving

car as a function of time, how can we compute the “instantaneous velocity ” of the car at a specific moment?

Possible approach: Compute the average velocity over a short period of time, and find the limit as that period approaches zero.

Limit of a Function at a Point In both problems above, we seek the

limit of some function (often, but not always, the function is in the form of a ratio) as we approach some point.

Definition: We say the limit as x approaches a of f (x) is a number L, writing limxa f (x) = L, if f ‘s values get closer and closer to L as x gets closer and closer to a.

Some Examples of Limits Some limits are obvious:

limx 3 x 2 = limt cos(t) =

But some limits aren’t: limt 0 sin(t) / t =

limh 0 ((3 + h)2 - 9) / h =

What was “problematic” about these two?

Clicker Question 1 What is limx 8 (ex – 8 + log2(x ))?

A. e + 2 B. 3 C. e + 3 D. 4 E. Does not exist

Clicker Question 2 What is limx 3 (x 2 – 9) / (x – 3) ?

A. 3 B. 6 C. 0 D. 1 E. Does not exist

Other Not Obvious Limits What is limx 4 (x 2 – 3x - 4)/(x – 4) ? What is limx 2 3/(x - 2)2 ? What is limx 1/x ? Note that in the last two examples, we are

allowing the idea of infinity to be involved in limits, either as the answer (meaning the output keeps getting bigger and bigger) or as what x approaches (meaning x gets bigger and bigger).

Assignment for Monday Here we go with calculus! Read Section 2.2. In Section 2.2, do Exercises 1, 21,

and 23.