Here, you will learn definition of limit in calculus, left hand limit, right hand limit and fundamental theorem of limit.
Letโs begin โ
Definition of Limit in Calculus
Let f(x) be defined on an open interval about โaโ except possibly at โaโ itself. If f(x) gets arbitrarily close to L(a finite number) for all x sufficiently close to โaโ we say that f(x) approaches the limit L as x approaches โaโ and we write \(\displaystyle{\lim_{x \to a}}\) f(x) = L and say โthe limit of f(x), as x approaches a, equals Lโ.
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 โaโ 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 โ h).
Right hand limit
The value to which f(x) approaches, as tends to โaโ 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.ย Also it is necessary to note that if f(x) is defined only on one side of โx = aโ, one sided limits are good enough to establish the existence of limits, & if f(x) is defined on either side of โaโ 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 itโs 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)ย Sum rule : \(\displaystyle{\lim_{x \to a}}\) {f(x) + g(x)} = l + m
(b)ย Difference rule : \(\displaystyle{\lim_{x \to a}}\) {f(x) โ g(x)} = l โ m
(c)ย Product rule : \(\displaystyle{\lim_{x \to a}}\) f(x).g(x) = l.m
(d)ย Quotient rule : \(\displaystyle{\lim_{x \to a}}\) \(f(x)\over g(x)\) = \(l\over m\)
(e)ย Constant multiple rule : \(\displaystyle{\lim_{x \to a}}\) kf(x) = k \(\displaystyle{\lim_{x \to a}}\) f(x)
(f)ย 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)ย \(\displaystyle{\lim_{x \to a}}\) 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).
