Limit Formulas

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1. Indeterminate form

0 × ∞, 0°, 1^∞, ∞ – ∞, ∞/∞, ∞°, 0/0.

2. Properties of Logarithms

Let M and N arbitrary positive number such that a > 0, a ≠ 1, b > 0, b ≠ 1 then

• log_a MN = log_a M + log_a N
• log_a\(\frac{M}{N}\) = log_a M – log_a N
• log_a N^α = α log_a N (α any real no.)
• log_a^βN^α = \(\frac{α}{β}\) log_aN (α ≠ 0, β ≠ 0)
• log_aN = \(\frac{\log _{b} N}{\log _{b} a}\)
• log_ba.log_ab = 1 ⇒ log_ba \(=\frac{1}{\log _{a} b}\)
• e^{ln ax} = a^x

3. Logarithmic Inequality

Let a is real number such that

• For a > 1 the inequality log_a x > log_a y & x > y are equivalent.
• If a> 1 then log_ax < α ⇒ 0 < x < a^α
• If a > l then log_ax > α ⇒ x > a^α
• For 0 < a < 1 the inequality 0 < x < y & log_a x > log_a y are equivalent
• If 0 < a < 1 then log_a x < α ⇒ x > a^α

4. Important discussion

(i) Given a number N, Logarithms can be expressed as

• The mantissa part of log of a number is always kept positive.
• If the characteristics of log₁₀ N be n then the number of digits in N is (n + 1)
• If the characteristics of log₁₀ N be (-n) then there exists (n – 1) number of zeros after decimal point of N.

(ii) If the no. & the base are on the same side of the unity, then the logarithm is positive; and If the no. and the base are on different side of unity. Then the logarithm is negative.

5. Limits of a function

\(\lim _{x \rightarrow a}\) f(x) = l
For finding right hand limit of the function we write (x + h) in place of x while for left hand limit we write (x – h) in place of x.

6. Existence of limit

Let f be a function in “x”. If for every positive number ∈, how ever small it may be there, ∃ a positive number δ such that whenever 0 < |x – a| < δ we have |f(x) – l| < ∈ then we say, f(x) tends to limit “l” as x tends to “a” and we say \(\lim _{x \rightarrow a}\) f(x) = l
Also in then neighbourhood of “a”
Right hand limit = Left hand limit or
\(\lim _{x \rightarrow a^{+}}\) f(x) = \(\lim _{x \rightarrow a^{-}}\) f(x)
Then we say limit of function “f” exist as x → a
How to find R.H.L. and L.H.L
For R.H.L.: → Replace x by “a + h” and write R.H.L. = \(\lim _{h \rightarrow 0}\) f(a + h)
For L.H.L.: → Replace x by “a – h” and write L.H.L. = \(\lim _{h \rightarrow 0}\) f(a – h)
Also R.H.L. can be written as \(\lim _{x \rightarrow a^{+}}\) f(x) or f(a + 0)
and L.H.L. can be written as \(\lim _{x \rightarrow a^{-}}\) f(x) or f(a – 0)

7. Methods of evaluation of limits

(A) When x → ∞
In this case expression should be expressed as a function 1/x and then after removing indeterminant form, (If it is there) replace 1/x by 0.

(B) Factorisation method
If f(x) is of the form \(\frac{g(x)}{h(x)}\) and of indeterminate form then this form is removed by factorising g(x) and h(x) and cancel the common factors, then put the value of x.

(C) Rationalisation method
In this method we rationalise the factor containing the square root and simplify and we put the value of x.

(D) By using some standard expansion

• e^x = 1 + x + \(\frac{x^{2}}{2 !}+\frac{x^{3}}{3 !}\) + ….
• e^-x = 1 – x + \(\frac{x^{2}}{2 !}-\frac{x^{3}}{3 !}\) + ….
• log(1 +x) = x – \(\frac{x^{2}}{2}+\frac{x^{3}}{3}\) – ……..
• log(1 – x)= – x – \(\frac{x^{2}}{2}-\frac{x^{3}}{3}\) – …….
• a^x = 1 + (x log a) + \(\frac{(x \log a)^{2}}{2 !}+\frac{(x \log a)^{3}}{3 !}\) + …….
• sin x = x – \(\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}\) – ….
• cos x = 1 – \(\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}\) – ….
• tan x = x + \(\frac{x^{3}}{3}+\frac{2}{15}\) x⁵ + . . .
• sinh x= x + \(\frac{x^{3}}{3 !}+\frac{x^{5}}{5 !}\) + …
• cosh x = 1 + \(\frac{x^{2}}{2 !}+\frac{x^{4}}{4 !}\) + …
• tanh x = x – \(\frac{x^{3}}{3}\) + 2x⁵ – ……
• sin^-1x = x + \(\frac{x^{3}}{3 !}+\frac{9 x^{5}}{5 !}\) + ……..
• cos^-1x = \(\frac{\pi}{2}-\left(x+\frac{x^{3}}{3 !}+\frac{9 x^{5}}{5 !}+\ldots\right)\)
• tan^-1x = x – \(\frac{x^{3}}{3}+\frac{x^{5}}{5}-\frac{x^{7}}{7}\) + …
• (l +x)ⁿ = 1 + nx + \(\frac{n(n-1)}{2 !} x^{2}\) + …
• (1 + x)^1/x = e[1 – \(\frac{x}{2}+\frac{11}{24}\) x² -………..]

(E) ‘L’ Hospital rule
If \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}\) is of the form \(\frac{0}{0}\) or \(\frac{∞}{∞}\), then \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}\)

(F) By using some standard limits
(a) \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=\lim _{x \rightarrow 0} \frac{x}{\sin x}=1 ; \lim _{x \rightarrow 0} \sin x=0\)

(b) \(\lim _{x \rightarrow 0} \cos x=\lim _{x \rightarrow 0}\left(\frac{1}{\cos x}\right)=\) 1

(c) \(\lim _{x \rightarrow 0} \frac{\tan x}{x}=\lim _{x \rightarrow 0} \frac{x}{\tan x}=1 ; \lim _{x \rightarrow 0} \tan x=0\)

(d) \(\lim _{x \rightarrow 0} \frac{\sin ^{-1} x}{x}=\lim _{x \rightarrow 0} \frac{x}{\sin ^{-1} x}=1\)

(e) \(\lim _{x \rightarrow 0} \frac{\tan ^{-1} x}{x}=\lim _{x \rightarrow 0} \frac{x}{\tan ^{-1} x}=1\)

(f) \(\lim _{x \rightarrow \infty}\left(1+\frac{a}{x}\right)^{x}=\lim _{x \rightarrow 0}(1+a x)^{1 / x}=e^{a}\)

(g) \(\lim _{x \rightarrow 0} \frac{a^{x}-1}{x}\) = log_e a (a > 0)

(h) \(\lim _{x \rightarrow 0} \frac{e^{x}-1}{x}\) = 1

(i) \(\lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}\) = na^n-1

(j) \(\lim _{x \rightarrow 0} \frac{\log (1+x)}{x}\) = 1

(k) \(\lim _{x \rightarrow 0} \frac{(1+x)^{n}-1}{x}\) = n

(l) \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}=\lim _{x \rightarrow \infty} \frac{\cos x}{x}=0\)

(m) \(\lim _{x \rightarrow \infty} \frac{\sin (1 / x)}{(1 / x)}\) = 1

8. Theorems on limits

The following theorems are very helpful for evaluation of limits:

• \(\lim _{x \rightarrow a}\)[k f(x)] = k \(\lim _{x \rightarrow a}\) f(x), where k is a constant
• \(\lim _{x \rightarrow a}\) [f(x) + g(x)] = \(\lim _{x \rightarrow a}\) f(x) + \(\lim _{x \rightarrow a}\)g(x)
• \(\lim _{x \rightarrow a}\) [f(x) – g(x)] = \(\lim _{x \rightarrow a}\) f(x) – \(\lim _{x \rightarrow a}\) g(x)
• \(\lim _{x \rightarrow a}\)[f(x).g(x)] = \(\lim _{x \rightarrow a}\)f(x).\(\lim _{x \rightarrow a}\)g(x)
• \(\lim _{x \rightarrow a}\)[f(x)/g(x)] = [\(\lim _{x \rightarrow a}\)f(x)]/\([\lim _{x \rightarrow a}\)g(x)] provided \(\lim _{x \rightarrow a}\)g(x) ≠ 0
• \(\lim _{x \rightarrow a}\)f[g(x)] = f[\(\lim _{x \rightarrow a}\)g(x)]
• \(\lim _{x \rightarrow a}\)[f(x) + k] = \(\lim _{x \rightarrow a}\)f(x) + k where k is a constant
• \(\lim _{x \rightarrow a}\)log{f(x)} = log{\(\lim _{x \rightarrow a}\)f(x)}
  • The Logarithm of a number is unique i.e. No number can have two different log to a given base.
  • From the definition of the Logarithm of the number to a given base ‘a’.
    a^log_aN = N, a > 0, a ≠ 1, & N > 0 is known as the fundamental logarithmic identity.
  • log_e a = log₁₀ a.log_e 10 or log₁₀a = \(\frac{\log _{e} a}{\log _{e} 10}\) = 0.434 log_e a.
• \(\lim _{x \rightarrow a}\)[f(x)]^g(x) = {\(\lim _{x \rightarrow a}\)f(x)}^{\(\lim _{x \rightarrow a}\)g(x)}
• \(\lim _{x \rightarrow ±∞}\)f(x) = \(\lim _{x \rightarrow 0}\)f(1/x)

9. Some limits which do not exist

• \(\lim _{x \rightarrow 0}\left(\frac{1}{x}\right)\)
• \(\lim _{x \rightarrow 0} x^{1 / x}\)
• \(\lim _{x \rightarrow 0} \frac{|x|}{x}\)
• \(\lim _{x \rightarrow a} \frac{|x-a|}{x-a}\)
• \(\lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)\)
• \(\lim _{x \rightarrow 0} \cos \left(\frac{1}{x}\right)\)
• \(\lim _{x \rightarrow 0}\) e^1/x
• \(\lim _{x \rightarrow ∞}\) sin x
• \(\lim _{x \rightarrow ∞}\) cos x

10. If function takes any of the following form, \(\frac{0}{0}\), \(\frac{∞}{∞}\), then L’HOSPITAL’S RULE is applied
\(\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\lim _{x \rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}\)
NOTE: L’HOSPITAL’S RULE can be repeated required number of times in same Question.

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