Limits and Derivatives

Limits


  • Limit of a function: Let a function ‘f’ be defined at every point in the neighbourhood of ‘a’ (an open interval about a) except possibly at a. If as ‘x’ approaches closer and closer to ‘a’, but not equal to ‘a’. Then the value of the function f(x) approaches a real number ‘l’. The number ‘l’ is referred to as the limit of f(x) as ‘x’ tends to ‘a’ and we write it as \lim_{x \rightarrow a}\ f(x) = l
  • When f(x) approaches ‘l’ means the absolute difference between f(x) and ‘l’, i.e |f(x) – l| can be made as small as we please.
  • When the values of f(x) do not approach a single finite value as ‘x’ approaches ‘a’. We say that the limit does not exist.
  • A number is said to be a limiting value only if it is finite and real, otherwise we say that the limit does not exist.
  • Right hand limit is denoted as \lim_{x \rightarrow a^{+}}\ f(x) = l_{1} (or) f(a + 0) = l1 and left hand limit is denoted as \lim_{x \rightarrow a^{-}}\ f(x) = l_{2} (or) f(a – 0) = l2
  • For finding \lim_{x \rightarrow a}\ f(x), we study the behaviour of the function ‘f’ in the neighbourhood of ‘a’ and not at ‘a’. Thus, the function ‘f’ may or may not be defined at x = a
  • The indeterminate forms are ∞ - ∞, \tt \frac{\infty}{\infty}, 0 × ∞, 1∞, 0° and ∞°
  • \lim_{x \rightarrow a}\ \left[f\left(x\right)\pm\ g\left(x\right)\right]=\lim_{x \rightarrow a}\ f\left(x\right)\pm\ \lim_{x \rightarrow a}\ g\left(x\right)
  • \lim_{x \rightarrow a}\ k\cdot f\left(x\right)=k\cdot\lim_{x \rightarrow a}f\left(x\right)
  • \lim_{x \rightarrow a}\ f\left(x\right)\cdot g\left(x\right)=\lim_{x \rightarrow a}f\left(x\right)\ \lim_{x \rightarrow a}\cdot\ g\left(x\right)
  • \lim_{x \rightarrow a}\ \frac{f\left(x\right)}{g\left(x\right)}=\frac{\lim_{x \rightarrow a}\ f\left(x\right)}{\lim_{x \rightarrow a}\ g\left(x\right)}
  • \lim_{x \rightarrow a}\ fog\left(x\right)=f\left[\lim_{x \rightarrow a}\ g\left(x\right)\right]
  • \lim_{x \rightarrow a}\ log\ g\left(x\right)=\ log\left[\lim_{x \rightarrow a}\ g\left(x\right)\right]
  • \lim_{x \rightarrow a}\ e^{g\left(x\right)}=\ e^{\lim_{x \rightarrow a}g\left(x\right)}
  • \lim_{x \rightarrow a}\left[f\left(x\right)\right]^{n}=\left[\lim_{x \rightarrow a}f\left(x\right)\right]^{n}
  • If f, g and h are functions such that f(x) ≤ g(x) ≤ h(x) ∀ x in some neighbourhood of the point a (except possibly at x = a) and if \lim_{x \rightarrow a}\ f\left(x\right)=l\ = \lim_{x \rightarrow a}\ h\left(x\right),\ then\ = \lim_{x \rightarrow a}\ g\left(x\right)=l
  • Sandwich theorem helps in calculating the limits, when limits cannot be calculated using the usual method.
  • If \lim_{x \rightarrow a}\ f\left(x\right)=l,\ then\ \begin{vmatrix}\lim_{x \rightarrow a} f\left(x\right)\end{vmatrix}=\mid l\mid but converse may or may not be true.
  • \lim_{x \rightarrow a}\ \frac{x^{n}-a^{n}}{x-a}= n\cdot a^{n-1}
  • \lim_{x \rightarrow \propto}\ \frac{1}{x}= 0\ and\ \lim_{x \rightarrow \propto}\ \frac{1}{x^{p}}=0 if p > 0.
    Tricks:
  • \lim_{n \rightarrow \infty}\ a^{n}=\infty if a > 1
                                          = 1 if a = 1
                                          = 0 if -1 < a < 1
                                          = does not exist if a ≤ -1
  • \lim_{n \rightarrow \infty}\ \frac{a_{0}x^{p}+a_{1}x^{p-1}+---+a_{p-1}\cdot x+a_{p}}{b_{0}x^{q}+b_{1}x^{q-1}+---+b_{q-1}\cdot x+b_{q}}=\frac{a_{0}}{b_{0}} if p = q
                                                                = 0 if p < q
                                                                = ∞ if p > q.

Part1: View the Topic in this video From 00:40 To 55:22

Part2: View the Topic in this video From 00:40 To 21:00

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1. Fundamental theorems on limits: If f(x) and g(x) be two functions of x such that \lim_{x \rightarrow a}f(x) and \lim_{x \rightarrow a}g(x) both exist, then

    (i) \lim_{x \rightarrow a}[f(x)\pm g(x)]=\lim_{x \rightarrow a}f(x)\pm\lim_{x \rightarrow a}g(x)

    (ii) \lim_{x \rightarrow a}[kf(x)]=k\lim_{x \rightarrow a}f(x), where k is a fixed real number.

    (iii) \lim_{x \rightarrow a}[f(x) g(x)]=\lim_{x \rightarrow a}f(x)\lim_{x \rightarrow a}g(x)

    (iv) \lim_{x \rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x \rightarrow a}f(x)}{\lim_{x \rightarrow a}g(x)},\lim_{x \rightarrow a}g(x)\neq 0

    (v) \lim_{x \rightarrow a}[f(x)]^{g(x)}=\left[\lim_{x \rightarrow a}f(x)\right]^{\lim_{x \rightarrow a}g(x)}=e^{\lim_{x \rightarrow a}\left\{f(x)-1\right\}g(x)}

    (vi) \lim_{x \rightarrow a}g[f(x)]=g\left[\lim_{x \rightarrow a}f(x)\right]

    (vii) \lim_{x \rightarrow a} \log f(x)=\log\left[\lim_{x \rightarrow a}f(x)\right], provided \lim_{x \rightarrow a} f(x)>0.

    (viii) \lim_{x \rightarrow a} e^{f(x)}=e^{\lim_{x \rightarrow a} f(x)}

2. Exponential limits:

    (i) \lim_{x \rightarrow 0} \frac{e^{x}-1}{x}=1

    (ii) \lim_{x \rightarrow 0} \frac{a^{x}-1}{x}=\log_{e}{a}

    (iii) \lim_{x \rightarrow 0} \frac{e^{\lambda x}-1}{x}=\lambda, where (λ ≠ 0).

3. Logarithmic limits:

    (i) \lim_{x \rightarrow 0} \frac{\log(1+x)}{x}=1

    (ii) \lim_{x \rightarrow e} \log_{e}x=1

    (iii) \lim_{x \rightarrow 0} \frac{\log_e(1-x)}{x}=-1

    (iv) \lim_{x \rightarrow 0} \frac{\log_a(1+x)}{x}=\log_{a}{e}, a > 0, \neq 1

4. Based on the form 1∞:

   (i) \lim_{x \rightarrow 0}(1+x)^{\frac{1}{x}}=e

   (ii) \lim_{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^x=e

   (iii) \lim_{x \rightarrow 0}(1+\lambda x)^{\frac{1}{x}}=e^{\lambda}

   (iv) \lim_{x \rightarrow \infty}\left(1+\frac{\lambda}{x}\right)^x=e^{\lambda}

5. (i) \lim_{x \rightarrow a}\frac{x^{a}-a^{x}}{x^{x}-a^{a}}=\frac{1-\log a}{1+\log a}

    (ii) \lim_{x \rightarrow 0}\frac{(1+x)^{m}-1}{(1+x)^{n}-1}=\frac{m}{n}

    (iii) \lim_{x \rightarrow 0}\frac{(1+bx)^{m}-1}{(1+ax)^{n}-1}=\frac{mb}{na}

    (iv) \lim_{x \rightarrow 0}(1+ax)^{b/x}=\lim_{x \rightarrow \infty}\left[1+\frac{a}{x}\right]^{bx}=e^{ab}

    (v) \lim_{n \rightarrow \infty}(x^{n}+y^{n})^{\frac{1}{n}}=y, (0 < x < y)

    (vi) \lim_{x \rightarrow \infty}\left(\frac{x\pm a}{x\pm b}\right)^{x+c}=e^{(a\mp b)}