Limits of piecewise functions

Limits of piecewise functions can differ from the direction we’re approaching them from

For example

f(x) = for 0 < x 4 for x > 4

lim f(x) = as it’s approaching from the right side (4.001, 4.01) which is greater than 4 x -> 4

lim f(x) = = = 2 as it’s approaching from the left side (3.99, 3.9) which is less than 4 x -> 4

If the limit approaching from left and the right side is the same then the limit exists

otherwise it doesn’t

Example:

lim f(x) = 2 x -> 4

Both left and right side equal 2 the limit as x -> 4 is 2!