A left hand limit of a function f(x) at a point x = a is the value that f(x) approaches as x gets closer and closer to a from the left side, i.e., from values less than a. We write it as lim x → a − f (x) = L, where L is the left hand limit.
For example, consider the function f(x) = |x| / x. If we want to find the left hand limit of f(x) at x = 0, we can plug in values of x that are slightly less than 0, such as -0.1, -0.01, -0.001, etc. We see that f(x) gets closer and closer to -1 as x approaches 0 from the left. Therefore, lim x → 0 − f (x) = -1.
However, not every function has a left hand limit at every point. For instance, if the function has a vertical asymptote or a jump discontinuity at x = a, then the left hand limit does not exist.
For example, consider the function g(x) = 1 / (x - 2). If we try to find the left hand limit of g(x) at x = 2, we see that g(x) becomes very large and negative as x gets closer and closer to 2 from the left.
There is no finite value that g(x) approaches, so lim x → 2 − g (x) does not exist.
I hope this helps you understand the concept of a left hand limit in calculus.