Limit is an important instrument that helps us understand ideas in the realm of Calculus. In this section, we will learn how to find the limit of a function graphically using one-sided limits and two-sided limits.

DEFINITION:

left-hand limit: $\lim_{x \to a^-} f(x) = L$

We say "the limit of f(x), as x approaches a from the negative direction, equals L".

It means that the value of f(x) becomes closer and closer to L as x approaches a from the left, but x is not equal to a.

DEFINITION:

right-hand limit: $\lim_{x \to a^+} f(x) = L$

We say "the limit of f(x), as x approaches a from the positive direction, equals L".

It means that the value of f(x) becomes closer and closer to L as x approaches a from the right, but x is not equal to a.

DEFINITION:

$\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^+} f(x) = L$ and $\lim_{x \to a^-} f(x) = L$

left-hand limit: $\lim_{x \to a^-} f(x) = L$

We say "the limit of f(x), as x approaches a from the negative direction, equals L".

It means that the value of f(x) becomes closer and closer to L as x approaches a from the left, but x is not equal to a.

DEFINITION:

right-hand limit: $\lim_{x \to a^+} f(x) = L$

We say "the limit of f(x), as x approaches a from the positive direction, equals L".

It means that the value of f(x) becomes closer and closer to L as x approaches a from the right, but x is not equal to a.

DEFINITION:

$\lim_{x \to a} f(x) = L$ if and only if $\lim_{x \to a^+} f(x) = L$ and $\lim_{x \to a^-} f(x) = L$