1 1 limits of a function

1.1 Limit of a Function Chapter Dr Tran Huong Lan LIMIT=TENDENCY First example x x2 f ( x)  ? x 4 x2 as We create a table of values for x close to x x x2 x2  1.9 1.99 1.99 2.001 2.01 2.1 0.746 0.749 0.7499 || 0.75 0.750 0.751 0.756 After this lesson, you will know • • • • • Definition of limits How to estimate a limit When a limit exists and does not exist How to interpret limits How to estimate limits by graphing Informal definition of limit Example We create a table of values for x close to x sin x cos x -0.1 -0.01 -0.001 -0.01 -0.001 -0.0001 0 0.001 0.01 0.0001 0.001 0.1 0.01 0.995 0.99995 0.9999995 0.9999995 0.99995 0.995 Note that the limit at IS NOT the value of the function at One-sided limits • Right-hand limit lim f ( x)  L (x  c) x c • Left-hand limit lim f ( x)  L (x  c)  x c  lim h( x)  1 x 3 lim h( x)  x 3 lim h( x) does not exist x 3 One-sided limit theorm The two-sided limit lim f ( x)  L if and only if x c lim f ( x)  L  lim f ( x) x c x c Example: find limits from graphs Estimate the following limits lim f ( x)  x 0 lim g ( x) x 1 does not exist lim h( x)  2 x 1 Example: find limits from graphs Estimate the following limits when x  y sin x lim 1 x 0 x sin x x y x2 lim x 0 x does not exist y  sin limsin x 0 x does not exist x Infinite limits Formal definition of a limit Epsilon-delta definition of a limit L is the LIMIT The interval always contains c no matter how small  is Epsilon-delta definition of a limit L is the LIMIT The interval always contains c no matter how small  is Epsilon-delta definition of a limit NOT THE LIMIT L* The interval does not contain c ! ... Definition of limits How to estimate a limit When a limit exists and does not exist How to interpret limits How to estimate limits by graphing Informal definition of limit Example We create a table of. ..LIMIT=TENDENCY First example x x2 f ( x)  ? x 4 x2 as We create a table of values for x close to x x x2 x2  1. 9 1. 99 1. 99 2.0 01 2. 01 2 .1 0.746 0.749 0.7499 || 0.75 0.750 0.7 51 0.756 After this lesson,... table of values for x close to x sin x cos x -0 .1 -0. 01 -0.0 01 -0. 01 -0.0 01 -0.00 01 0 0.0 01 0. 01 0.00 01 0.0 01 0 .1 0. 01 0.995 0.99995 0.9999995 0.9999995 0.99995 0.995 Note that the limit at IS NOT