Limits An Introduction To: Changing Rates Secant and Tangent Lines &

Suppose you drive 200 miles, and it takes you 4 hours. What is your average speed? Your average speed is: If you look at your speedometer during this trip, it might read 65 mph. This is your instantaneous speed.

A rock falls from a high cliff. The change in position of the rock is given by: What is your average speed after 2 seconds: average speed: What is the instantaneous speed at 2 seconds?

The tangent line problem secant line (secant means to cut) (c, f(c)) (c, f(c)) is the point of tangency and f(c+ ) – f(c) x is a second point on the graph of f.

Tangent and secant lines  

Limit of a Function Limit of a Function Let f be a function and let a and L be real numbers. L is the limit of f(x) as x approaches a, written if the following conditions are met. As x assumes values closer and closer (but not equal ) to a on both sides of a, the corresponding values of f(x) get closer and closer (and are perhaps equal) to L. The value of f(x) can be made as close to L as desired by taking values of x arbitrarily close to a.

Limits We can find limits: graphically, numerically, and algebraically. The next slide finds the limit graphically.

Limit of a Function  

Finding limits numerically  

Limit of a Function--numerically Values of may be computed near x = 2 x approaches 2   f(x) approaches 4 x 1.9 1.99 1.999 2.001 2.01 2.1 f(x) 3.9 3.99 3.999 4.001 4.01 4.1

Limit of a Function The values of f(x) get closer and closer to 4 as x gets closer and closer to 2. We say that “the limit of as x approaches 2 equals 4” and write

Finding the Limit of a Polynomial Function (graphically) Solution or from a graph of f(x). We see that

Finding the Limit of a Polynomial Function Example Find Solution The behavior of near x = 1 can be determined from a table of values, x approaches 1   f(x) approaches 2 x .9 .99 .999 1.001 1.01 1.1 f(x) 2.11 2.0101 2.001 1.999 1.9901 1.91

Finding the Limit of a Polynomial Function Example Find where Solution Create a graph and table.

Finding the Limit of a Polynomial Function Solution x approaches 3   f(x) approaches 7 Therefore x 2.9 2.99 2.999 3.001 3.01 3.1 f(x) 6.8 6.98 6.998 7.004 7.04 7.4

Limits That Do Not Exist If there is no single value that is approached by f(x) as x approaches a, we say that f(x) does not have a limit as x approaches a, or does not exist.

Limit formal definition (Calculus B_C)