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.