Limits and Derivatives

Limits of Trigonometric Functions

  • \lim_{x \rightarrow 0}\frac{\sin x}{x}=1
  • \lim_{x \rightarrow 0}\frac{\tan x}{x}=1
  • \lim_{x \rightarrow 0}\frac{\sin^{-1} x}{x}=1
  • \lim_{x \rightarrow 0}\frac{\tan^{-1} x}{x}=1
  • \sin x=x-\frac{x^{3}}{3!}+\frac{x^{5}}{5!}-\frac{x^{7}}{7!}+.....\infty\ ;\ x\ \epsilon\ R
  • \cos x=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^{6}}{6!}+.....\infty\ ;\ x\ \epsilon\ R
  • \tan x=x+\frac{x^{3}}{3}+\frac{2}{15}\ x^{5}+\frac{17}{315}\ x^{7}+\frac{62}{2835}\ x^{9}+.....\infty\ \mid x\mid<\frac{\pi}{2}
  • \sec x=1+\frac{x^{2}}{2}+\frac{5}{24}\ x^{4}+\frac{61}{720}\ x^{6}+.......\infty\ 0<\mid x\mid<\frac{\pi}{2}
  • \\cosec\ x=\frac{1}{x}+\frac{x}{6}+\frac{7}{360} x^{3}+\frac{31}{15120}\ x^{5}+.....\infty;\ 0<\mid x\mid<{\pi}
  • \cot\ x=\frac{1}{x}-\frac{x}{3}-\frac{x^{3}}{45}-\frac{2}{945}\ x^{5}-....\infty;\ 0<\mid x\mid<{\pi}
  • \sin^{-1}x=x+\frac{1}{2}\cdot \frac{x^{3}}{3}+\frac{1}{2}\cdot\frac{3}{4}\ \cdot \frac{x^{5}}{5}+\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\ \frac{x^{7}}{7}+......\infty;\ \mid x\mid<1
  • Exponential and logarithmic limits


    • \lim_{x \rightarrow 0}\frac{e^{x}-1}{x}=1
    • \lim_{x \rightarrow 0}\frac{a^{x}-1}{x}=\log_{e}{a};\ a > 0
    • \lim_{x \rightarrow 0}\frac{a^{x}-b^{x}}{x}=\log_{e}(\frac{a}{b});\ a,\ b > 0
    • \lim_{x \rightarrow 0}\frac{\left(1+x\right)^{n}-1}{x}=n
    • \lim_{x \rightarrow 0}\left(1+x\right)^{1/x}=\lim_{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}=e
    • \lim_{h \rightarrow 0}\left(1+ah\right)^{1/h}=e^{a}
    • \lim_{x \rightarrow \infty}\ \frac{\log{x}}{x^{m}}=0;\ (m>0)
    • \lim_{x \rightarrow0}\ \frac{\log_{a}{\left(1+x\right)}}{x}=\log_{a}{e},\ \left(a>0, a\neq1\right)
    • \lim_{x \rightarrow\infty}\ \left(1+\frac{a}{x}\right)^{x}=e^{a}
    • \lim_{x \rightarrow\infty}\left[1+\frac{1}{f\left(x\right)}\right]^{f\left(x\right)}=e, where f(x) → ∞ as x → ∞
    • \lim_{x \rightarrow a}\left[1+f\left(x\right)\right]^{\frac{1}{f\left(x\right)}}=e
    • \tt e^{x}=\ 1+\frac{x}{1!}+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+.....\ to\ \infty
    • \tt e^{-x}=\ 1-\frac{x}{1!}+\frac{x^{2}}{2!}-\frac{x^{3}}{3!}+..... \ to\ \infty
    • \tt \log_{e}{\left(1+x\right)}=x-\frac{x^{2}}{2}+\frac{x^{3}}{3}.... \ to\ \infty,\ -1<x\leq1
    • \tt \log_{e}{\left(1-x\right)}=-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}..... \ to\ \infty,\ -1\leq x<1
    • \tt a^{x}=e^{x\cdot\log{a}}=1+x\log a+\frac{\left(x\cdot log\ a\right)^{2}}{2!}+..... \ to\ \infty
    • If \tt \lim_{x \rightarrow a}f\left(x\right)=0\ and\ \lim_{x \rightarrow a}\ g\left(x\right)=0,\ then\ \lim_{x \rightarrow a}\ \frac{f\left(x\right)}{g\left(x\right)}=\ \lim_{x \rightarrow a}\ \frac{f'\left(x\right)}{g'\left(x\right)}, provided the limit on the R.H.S exists. This is L’HOSPITAL’s RULE


    • If \tt \lim_{x \rightarrow a}\ f\left(x\right)=A>0\ and\ \lim_{x \rightarrow a}\ g\left(x\right)=B\ then\ \lim_{x \rightarrow a}\left[f\left(x\right)\right]^{g\left(x\right)}= A^{B}
    • \tt \lim_{x \rightarrow a}\ f\left(x\right)=1\ and\ \lim_{x \rightarrow a}\ g\left(x\right)=\infty\ then\ \lim_{x \rightarrow a}\ \left[f\left(x\right)\right]^{g\left(x\right)}=\ e^{\lim_{x \rightarrow a}g\left(x\right)\left[f\left(x\right)-1\right]}
    • L HOSPITAL’s Rule is applicable only when \tt \frac{f\left(x\right)}{g\left(x\right)} becomes of the form \tt \frac{0}{0}\ or\ \frac{\infty}{\infty}
    • If the form is not \tt \frac{0}{0}\ or\ \frac{\infty}{\infty}, simplify till to get this form
    • For applying L HOSPITAL’s rule differentiate the numerator and denominator separately

Part1: View the Topic in this video From 21:01 To 52:05

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1. Trigonometric limits :

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

    (ii) \lim_{x \rightarrow 0}\frac{\tan x}{x}=1=\lim_{x \rightarrow 0}\frac{x}{\tan x}

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

    (iv) \lim_{x \rightarrow 0}\frac{\tan^{-1} x}{x}=1=\lim_{x \rightarrow 0}\frac{x}{\tan^{-1} x}

    (v) \lim_{x \rightarrow 0}\frac{\sin x^{0} }{x}=\frac{\pi}{180^{o}}

    (vi) \lim_{x \rightarrow 0}\cos x = 1

    (vii) \lim_{x \rightarrow a}\frac{\sin(x-a)}{x-a} = 1

    (viii) \lim_{x \rightarrow a}\frac{\tan(x-a)}{x-a} = 1

    (ix) \lim_{x \rightarrow a}\sin^{-1}x=\sin^{-1}a, |a| \leq 1

    (x) \lim_{x \rightarrow a}\cos^{-1}x=\cos^{-1}a, |a| \leq 1

    (xi) \lim_{x \rightarrow a}\tan^{-1}x=\tan^{-1}a, -\infty < a < \infty

    (xii) \lim_{x \rightarrow \infty}\frac{\sin x}{x}=\lim_{x \rightarrow \infty}\frac{\cos x}{x}=0

    (xiii) \lim_{x \rightarrow \infty}\frac{\sin \frac{1}{x}}{\frac{1}{x}}=1

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

2. Trigonometric limits results :

    (i) \lim_{x \rightarrow 0}\frac{1-\cos m x}{1-\cos nx}=\frac{m^{2}}{n^{2}}

    (ii) \lim_{x \rightarrow 0}\frac{\cos ax-\cos b x}{\cos cx -\cos dx}=\frac{a^{2}-b^{2}}{c^{2}-d^{2}}

    (iii) \lim_{x \rightarrow 0}\frac{\cos mx-\cos n x}{x^{2}}=\frac{n^{2}-m^{2}}{2}

    (iv) \lim_{x \rightarrow 0}\frac{\sin^{p} mx}{^{p}nx}=\left(\frac{m}{n}\right)^{p}

    (v) \lim_{x \rightarrow 0}\frac{\tan^{p} mx}{tan^{p} \ nx}=\left(\frac{m}{n}\right)^{p}