Definition of Limit in Calculus – Theorem of Limit

Here, you will learn definition of limit in calculus, left hand limit, right hand limit and fundamental theorem of limit.

Let’s begin –

Definition of Limit in Calculus

Let f(x) be defined on an open interval about ‘a’ except possibly at ‘a’ itself. If f(x) gets arbitrarily close to L(a finite number) for all x sufficiently close to ‘a’ we say that f(x) approaches the limit L as x approaches ‘a’ and we write \(\displaystyle{\lim_{x \to a}}\) f(x) = L and say “the limit of f(x), as x approaches a, equals L”.

This implies if we can make the value of f(x) arbitrarily close to L(as close to L as we like) by taking x to be sufficiently close to a(on either side of a) but not equal to a.

Left hand limit and Right hand limit of a function

Left hand limit

The value to which f(x) approaches, as tends to ‘a’ from the left hand side (x \(\rightarrow\) \(a^{-}\)) is called left hand limit of f(x) at x = a.

Symbolically, LHL = \(\displaystyle{\lim_{x \to a^-}}\) f(x) = \(\displaystyle{\lim_{h \to 0}}\) f(a – h).

Right hand limit

The value to which f(x) approaches, as tends to ‘a’ from the right hand side (x \(\rightarrow\) \(a^{+}\)) is called right hand limit of f(x) at x = a.

Symbolically, RHL = \(\displaystyle{\lim_{x \to a^+}}\) f(x) = \(\displaystyle{\lim_{h \to 0}}\) f(a + h).

Limit of a function f(x) is said to exist as, x \(\rightarrow\) a when \(\displaystyle{\lim_{x \to a^-}}\) f(x) = \(\displaystyle{\lim_{x \to a^+}}\) f(x) = Finite quantity

Note :

In \(\displaystyle{\lim_{x \to a}}\) f(x), x \(\rightarrow\) a necessarily implies x \(\ne\) a. This is while evaluating limit at x = a, we are not concerned with the value of the function at x = a. In fact the function may or may not be defined at x = a.  Also it is necessary to note that if f(x) is defined only on one side of ‘x = a’, one sided limits are good enough to establish the existence of limits, & if f(x) is defined on either side of ‘a’ both sided limits are to be considered.

As in \(\displaystyle{\lim_{x \to a}}\) \(\cos^{-1}x\) = 0, though f(x) is not defined for x > 1, even in it’s immediate vicinity.

Fundamental theorem of limit

Let \(\displaystyle{\lim_{x \to a}}\) f(x) = l

\(\displaystyle{\lim_{x \to a}}\) g(x) = m. If l & m exist finitely then :

(a)  Sum rule : \(\displaystyle{\lim_{x \to a}}\) {f(x) + g(x)} = l + m

(b)  Difference rule : \(\displaystyle{\lim_{x \to a}}\) {f(x) – g(x)} = l – m

(c)  Product rule : \(\displaystyle{\lim_{x \to a}}\) f(x).g(x) = l.m

(d)  Quotient rule : \(\displaystyle{\lim_{x \to a}}\) \(f(x)\over g(x)\) = \(l\over m\)

(e)  Constant multiple rule : \(\displaystyle{\lim_{x \to a}}\) kf(x) = k \(\displaystyle{\lim_{x \to a}}\) f(x)

(f)  Power rule : If m and n are integers then \(\displaystyle{\lim_{x \to a}}\) \([f(x)]^{m/n}\) = \(l^{m/n}\) provided
\(l^{m/n}\) is a real number.

(g)  \(\displaystyle{\lim_{x \to a}}\) f[g(x)] = f(\(\displaystyle{\lim_{x \to a}}\) g(x)) = f(m); provided f(x) is continuous at x = m.

For Example :\(\displaystyle{\lim_{x \to a}}\) ln(g(x)) = ln[\(\displaystyle{\lim_{x \to a}}\) g(x)] = ln(m); provided lnx is continuous at x = m, m = \(\displaystyle{\lim_{x \to a}}\) g(x).

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