Calculus Solutions, Important Examples, Formulas and Videos

Algebra of limit

Theorem 1: Let f and g be two functions such that both \lim_{x\rightarrow a}f(x)\: and\: \lim_{x\rightarrow 0}g(x) exist then.

  • Limit of sum of two functions is sum of the limits of the function :
    \lim_{x\rightarrow a}[f(x)+g(x)]=\lim_{x\rightarrow a}f(x)+\lim_{x\rightarrow 0}g(x)
  • Limit of difference of two functions is difference of the limits of the functions, i.e
    \lim_{x\rightarrow a}[f(x)-g(x)]=\lim_{x\rightarrow a}f(x)-\lim_{x\rightarrow 0}g(x)
  • Limit of product of two functions is product of the limits of the functions, i.e
    \lim_{x\rightarrow a}[f(x)\cdot g(x)]=\lim_{x\rightarrow a}f(x)\cdot\lim_{x\rightarrow 0}g(x)
  • Limit of quotient of two functions is quotient of the limits of the functions (whenever the denominator is non zero) , i .e
    \lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow 0}f(x)}{\lim_{x\rightarrow a}g(x)}

Note: In particular as case of (iii), when g is the constant function such that g(x) = λ , for some real number λ ,

we have \lim_{x\rightarrow a}[(\lambda ,f)(x)]=\lambda \lim_{x\rightarrow a}f(x)

Limits of polynomials and rational function:

A function f is said to be a polynomial function if f(x) is zero function or if f(x)=a_{0}+a_{1}x+a_{2}x^{2}+......+a_{n}x^{n} , where a_{1} are real numbers such that a n ≠ 0 for some natural numbers n.

We know that \lim_{x\rightarrow a}=a Hence \lim_{x\rightarrow a}=x^{2}=\lim_{x\rightarrow a}(x\cdot x)=\lim_{x\rightarrow a}x\cdot \lim_{x\rightarrow a}x=a\cdot a=a^{2}
Or \lim_{x\rightarrow a}x^{n}=a^{n}

A function f is said to be a rational function if f(x)=\frac{g(x)}{h(x)} are polynomial such that  h(x)\neq 0 Then ,
\lim_{x\rightarrow a}\frac{g(x)}{h(x)}=\frac{\lim_{x\rightarrow a}g(x)}{\lim_{x\rightarrow a}f(x)}= \frac{g(a)}{h(a)}
Limit of trigonometric function : Let f and g be two real valued functions with the same domain such that f (x) ≤ g(x) for all x in the domain of definition, For some a, if both \lim_{x\rightarrow o}f(x)\: and\: g(x)\: exist\: then\: \lim_{x\rightarrow a}f(x)\leq \lim_{x\rightarrow 0}g(x) . in fig 13.8

Theorem:
The following are two important limits

  • \lim_{x\rightarrow 0}\frac{\sin x}{x}=1
  • \lim_{x\rightarrow 0}\frac{1-\cos x}{x}=0

Example: Evaluate \lim_{x\rightarrow 0}\frac{\sin4 x}{2x}

Solution: \lim_{x\rightarrow 0}\frac{\sin4 x}{2x}=\lim_{x\rightarrow 0}\left [ \frac{sin4x}{4x}\cdot \frac{2x}{sin2x}\cdot 2 \right ]= 4\lim_{x\rightarrow 0}\left [ \frac{\sin4x}{4x}\right ]\div \left [ \frac{\sin 2x}{2x} \right ]= 2\lim_{4x\rightarrow 0}\left [ \frac{\sin4x}{4x}\right ]\div \lim_{2x\rightarrow 0}\left [ \frac{\sin 2x}{2x} \right ]=2\times 1\times 1=2\left (as\: x\rightarrow 0,4x\rightarrow 0,\: and\: 2x\rightarrow 0 \right )

Derivatives: uppose f is a real valued function and a is a point in its domain of definition. The derivative of f at a is defined by \lim_{h\rightarrow 0}\frac{f(a+h)-f(a)}{h}
provided this limit exists. Derivative of f (x)at a is denoted by f ′(a). observe that f′ (a) quantifies the change in f(x) at a with respect to x.

Example: Find the derivative at x = 2 of the function f (x) = 3x.

Solution: f^{'}(2)=\lim_{h\rightarrow 0}\frac{f(2+h)-h}{h}=\lim_{h\rightarrow 0}=\frac{3(2+h)-3(2)}{h}=\lim_{h\rightarrow 0}\frac{6+3h-6}{h}=\lim_{h\rightarrow 0}\frac{3h}{h}=3 The derivative of the function 3x at x = 2 is 3 .
Algebra of derivative of functions: Since the very definition of derivatives involve limits in a rather direct fashion, we expect the rules for derivatives to follow closely that of limits. We collect these in the following theorem .

 

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