Consider the function f defined by the formula f(x) = sin(x)/x.

Obviously, this formula doesn't make sense when x = 0, so we can't use this formula by itself to define a function which is continuous everywhere. We'd like to define a function F so that F(x) = f(x) for all nonzero values of x, and such that F is continuous at 0.

The value F(0) is called the limit of f(x) as x approaches 0, usually written 

To find the limit graphically, graph f and use the ZOOM menu to focus your attention on the portion of the graph near x = 0.

To find the limit numerically, evaluate y1(x) for a variety of values of x which are close to 0. One way to automate this process is to visit the TABLE screen. If you adjust the TABLE SETUP menu, you can display several useful values of x and y1(x) at once.

Your calculator will also find the limit for you. Enter limit(y1(x),x,0) on the command line of the HOME screen.
 
 

Exercises: Find the indicated limits using all three methods. Not every limit actually exists.

• Find the limit as x approaches 0 of (e^x -1)/x. (If you just use the letter e instead of the exponentiation key, you'll get an answer with ln(e) in it. That's a good answer, but you'll need to know that ln(e) = 1.)

 
• Find the limit as x approaches 0 of (sin(x) - x)/ (x(cos(x) - 1)). Be careful entering the formula; you'll need a multiplication symbol between the x and the (cos( in the denominator.)

 
• Find the limit as x approaches 0 of cos(x)/x. What goes wrong here?

 
• Find the limit as x approaches 1 of (x³ - 1)/(x - 1). The calculator tends to make this too easy.

 
• Find the limit as x approaches 1 of (x³ - 1)/(sqrt(x) - 1). (The square root key is 2nd multiply.)

 
• Find the limit as x approaches infinity of x⁴/e^x. What numbers are "close to" infinity? What part of the graph do you want to look at?