Limits and Derivatives - Revision Notes

 CBSE Class 11 Mathematics

Revision Notes

Chapter-13

LIMITS AND DERIVATIVES

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1. Limits
2. Derivatives
3. Miscellaneous Questions

Meaning of x→a$x\to a$ or "x$x$ tends to a$a$" or "x$x$ approaches a$a$", x$x$ is a variable. The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the righthand limit. It can be changed so that its value comes nearer and nearer to a$a$.  0<|x−a|<δ$0<\left|x-a\right|<\delta$

(i) x≠a$x\ne a$,      (ii) |x−a|$\left|x-a\right|$ becomes smaller and smaller as we please.

Neighbourhood: The set of all real numbers lying between a−δ$a-\delta$ and a+δ$a+\delta$ is called the neighbourhood of a$a$. Neighbourhood of a$a$ = (a−δ,a+δ),$\left(a-\delta ,a+\delta \right),$     x∈(a−δ,a+δ)$x\in \left(a-\delta ,a+\delta \right)$

• Limit of a function at a point is the common value of the left and right hand limits, if they coincide

Left hand limit of f$f$ at x=a$x=a$. When x$x$ approches a$a$ from left hand side of a$a$, the function f(x)$f(x)$ tends to l$l$ "a definite number". This definite number l$l$ is said to be the left hand limit of f$f$ at x=a$x=a$.

Right hand limit of f$f$ at x=a$x=a$. When x$x$ approches a$a$ from right hand side of a$a$, the function f(x)$f(x)$ tends to l$l$ "a definite number". This definite number l$l$ is said to be the right hand limit of f$f$ at x=a$x=a$.

Therefore, if Left hand limit of f$f$ at x=a$x=a$ = Right hand limit of f$f$ at x=a$x=a$, then the limit of f(x)$f(x)$ at x=a$x=a$ exists.

• For function f and a real number a, limx→∞f(x)$\underset{x\to \mathrm{\infty }}{lim}f(x)$ and f (a) may not be same (Infact, one may be defined and not the other one).
• For functions f and g the following holds:.

limx→a[f(x)±g(x)]=$\underset{x\to a}{lim}\left[f(x)\pm g(x)\right]=$limx→af(x)±limx→ag(x)$\underset{x\to a}{lim}f(x)\pm \underset{x\to a}{lim}g(x)$

limx→a[f(x).g(x)]=$\underset{x\to a}{lim}\left[f(x).g(x)\right]=$limx→af(x).limx→ag(x)$\underset{x\to a}{lim}f(x).\underset{x\to a}{lim}g(x)$

limx→a[f(x)g(x)]=$\underset{x\to a}{lim}\left[\frac{f(x)}{g(x)}\right]=$limx→af(x)limx→∞g(x)$\frac{\underset{x\to a}{lim}f(x)}{\underset{x\to \mathrm{\infty }}{lim}g(x)}$

Following are some of the standard limits

limx→axn−anx−a=nan−1$\underset{x\to a}{lim}\frac{x^n-a^n}{x-a}=n{a}^{n-1}$

limx→0sinxx=1$\underset{x\to 0}{lim}\frac{\sin x}{x}=1$, limx→asin(x−a)x−a=1$\underset{x\to a}{lim}\frac{\sin \left(x-a\right)}{x-a}=1$

limx→01−cosxx=0$\underset{x\to 0}{lim}\frac{1-\cos x}{x}=0$

limx→0tanxx=1,$\underset{x\to 0}{lim}\frac{\tan x}{x}=1,$limx→atan(x−a)x−a=1$\underset{x\to a}{lim}\frac{\tan \left(x-a\right)}{x-a}=1$

limx→0sin−1xx=1,$\underset{x\to 0}{lim}\frac{{\sin }^{-1}x}{x}=1,$limx→0tan−1xx=1$\underset{x\to 0}{lim}\frac{{\tan }^{-1}x}{x}=1$

limx→0ax−1x=logea,$\underset{x\to 0}{lim}\frac{a^x-1}{x}={\log }_{e}a,$a>0,a≠1$a>0,a\ne 1$

Derivatives

The derivative of a function f at a is defined by

f′(a)=limh→0f(a+h)−f(a)h$f^{\prime }(a)=\underset{h\to 0}{lim}\frac{f(a+h)-f(a)}{h}$

Derivative of a function f at any point x is defined by

f′(x)=df(x)dx=limh→0f(x+h)−f(x)h$f^{\prime }(x)=\frac{df(x)}{dx}=\underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}$

For functions u and v the following holds:

(u±v)′=u′±v′$(u\pm v{)}^{\prime }=u^{\prime }\pm v^{\prime }$          

(uv)′=u′v+uv′$(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$            ⇒$\Rightarrow$          ddx(uv)=u.dvdx+v.dudx$\frac{d}{dx}\left(uv\right)=u.\frac{dv}{dx}+v.\frac{du}{dx}$

(uv)′=u′v−uv′v2${\left(\frac{u}{v}\right)}^{{}^{\prime }}=\frac{u^{\prime }v-u{v}^{\prime }}{v^2}$                 ⇒$\Rightarrow$          ddx(uv)=v.dudx−u.dvdxv2$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{v.\frac{du}{dx}-u.\frac{dv}{dx}}{v^2}$

provided all are defind.

Following are some of the standard derivatives

ddx(xn)=nxn−1$\frac{d}{dx}(x^n)=n{x}^{n-1}$

ddx(sinx)=cosx$\frac{d}{dx}(\sin x)=\cos x$

ddx(cosx)=−sinx$\frac{d}{dx}(\cos x)=-\sin x$

ddx(tanx)=sec2x$\frac{d}{dx}\left(\tan x\right)={\sec }^{2}x$

ddx(cotx)=−cosec2x$\frac{d}{dx}\left(\cot x\right)=-\cos e{c}^{2}x$

ddx(secx)=secx.tanx$\frac{d}{dx}\left(\sec x\right)=\sec x.\tan x$

ddx(cosecx)=−cosecx.cotx