Find a limit
See what value a function approaches as x gets close to a number, exactly or by testing nearby values.
A limit asks what y does as x gets very close to a value, even if the function is undefined exactly there. For example (x² − 9)/(x − 3) breaks at x = 3, but it heads towards 6.
A CAS calculator can find the limit exactly. Other calculators can still show the trend: plug in a number very close to the point.
Classic CAS · Exact limit
Start on the main screen: Press home to open the Home menu (Calculator, Graphs, Settings, and more). Choose Calculator, or press esc to go back to the calculation page.
Type limit( , the expression (x² − 9)/(x − 3), the variable and the value 3.
limit((x^2▶−9)÷(x−3),x,3)enter
You should seelimit((x^2−9)/(x−3),x,3) 6
Classic CAS · Limit at infinity
Start on the main screen: Press home to open the Home menu (Calculator, Graphs, Settings, and more). Choose Calculator, or press esc to go back to the calculation page.
Use the infinity key as the value. This is the famous limit that defines e.
limit((1+1÷n)^n▶,n,∞)enter
You should seee
Classic Graphing 84 · Test a nearby value
Start on the main screen: Press 2nd then mode (QUIT) to get back to the home screen. Pressing clear on a menu also backs out of it.
Type the expression with x replaced by 3.0001, a number just beyond 3. Use brackets around the top and bottom.
(3.0001x²−9)÷(3.0001−3)Press enter. The result is about 6, which is the limit.
enter
You should see6.0001
Tips
- Try a value on each side (2.9999 and 3.0001). If they head to different numbers, the limit does not exist.
- On the CAS, add 1 or -1 as a last argument for a one-sided limit.