Here, you will learn definition of limit in calculus, left hand limit, right hand limit and fundamental theorem of limit.<\/p>\n
Let’s begin –<\/p>\n
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”.<\/p>\n
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.<\/p>\n
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.<\/p>\n
\nSymbolically, LHL = \\(\\displaystyle{\\lim_{x \\to a^-}}\\) f(x) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) f(a – h).<\/p>\n<\/blockquote>\n
Right hand limit<\/strong><\/h4>\n
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.<\/p>\n
\nSymbolically, RHL = \\(\\displaystyle{\\lim_{x \\to a^+}}\\) f(x) = \\(\\displaystyle{\\lim_{h \\to 0}}\\) f(a + h).<\/p>\n
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<\/p>\n<\/blockquote>\n
Note :<\/strong><\/p>\n
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.\u00a0 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.<\/p>\n
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.<\/p>\n
Fundamental theorem of limit<\/h2>\n
Let \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x) = l<\/p>\n
\\(\\displaystyle{\\lim_{x \\to a}}\\) g(x) = m. If l & m exist finitely then :<\/p>\n
(a)\u00a0 Sum rule : \\(\\displaystyle{\\lim_{x \\to a}}\\) {f(x) + g(x)} = l + m<\/p>\n
(b)\u00a0 Difference rule : \\(\\displaystyle{\\lim_{x \\to a}}\\) {f(x) – g(x)} = l – m<\/p>\n
(c)\u00a0 Product rule : \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x).g(x) = l.m<\/p>\n
(d)\u00a0 Quotient rule : \\(\\displaystyle{\\lim_{x \\to a}}\\) \\(f(x)\\over g(x)\\) = \\(l\\over m\\)<\/p>\n
(e)\u00a0 Constant multiple rule : \\(\\displaystyle{\\lim_{x \\to a}}\\) kf(x) = k \\(\\displaystyle{\\lim_{x \\to a}}\\) f(x)<\/p>\n
(f)\u00a0 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.<\/p>\n(g)\u00a0 \\(\\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.<\/p>\n\n\n
For Example :<\/span>\\(\\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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