## Limits formula – Derivation formulas

### Limits formula

Limits math is very important in calculus. It is one of the basic prerequisites to understand other concepts in Calculus such as continuity, differentiation, integration limit formula, etc. Most of the time, math limit formula are the representation of the behaviour of the function at a specific point. Hence, the concept of limits is used to analyze the function.

The mathematical concept of Limit of a topological net generalizes the limit of a sequence and hence relates limits math to the theory category. Integrals in general are classified into definite and indefinite integrals.

The upper and lower limits are specified in the case of definite integration limit formula. However, indefinite integration limit formula are defined without specified limits and hence have an arbitrary constant after integration.

### Properties of Limits:

Let a and k are real numbers, and f(x) and g(x) be any two functions

### 1. Constant times a function:

$$\lim_{x\to a}[k \cdot f(x)] = k \lim_{x\to a} f(x)$$

### Example:

$$Evalute : \ \lim_{x\to 3} (5x^2)$$
$$\lim_{x\to 3} (5x^2) = 5 \lim_{x\to 3} (x^2)$$
$$=5(3^2)$$
$$=45$$

### 2. Sum of functions

$$\lim_{x\to a} [f(x)+g(x)] = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)$$

### Example:

$$Evalute : \ \lim_{x\to 3} (2x+5)$$
$$\lim_{x\to 3} (2x+5) = \lim_{x\to 3} (2x) + \lim_{x\to 3} (5)$$
$$= 2 \lim_{x\to 3} (x) + \lim_{x\to 3} (5)$$
$$=2(3)+5$$
$$=11$$

### 3. Difference of functions

$$\lim_{x\to a} [f(x)-g(x)] = \lim_{x\to a} f(x) – \lim_{x\to a} g(x)$$

### 4. Product of functions

$$\lim_{x\to a} [f(x) \cdot g(x)] = \lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x)$$

### 5. Quotient of functions

$$\lim_{x\to a} \frac {f(x)}{g(x)} = \frac {{ \lim\limits_{x \to a} f(x) }} {{ \lim\limits_{x \to a} g(x) }} = \frac AB$$

### 6. Function raised to an exponent

$$\lim_{x\to a} [f(x)]^n = \left[ \lim_{x\to \infty} f(x) \right]^n$$

### Example:

$$Evalute : \ \lim_{x\to 2} (3x+1)^5$$
$$\lim_{x\to 2} (3x+1)^5 = \left( \lim_{x\to 2} (3x+1) \right) ^5$$
$$=(3(2)+1)^5$$
$$=7^5$$
$$=16,807$$

### 7. nth root of a function, where n is a positive integer

$$\lim_{x\to a} {\sqrt[n] {f(x)}} = \sqrt[n] { \lim_{x\to a} {[f(x)]}}$$

### Limits of Trigonometry functions:

$$1. \ \lim_{x\to 0} {sin \ x} = 0$$
$$2. \ \lim_{x\to 0} {cos \ x} = 1$$
$$3. \ \lim_{x\to 0} { \frac {sin \ x}{x}} = 1$$
$$4. \ \lim_{x\to 0} { \frac {tan \ x}{x}} = 1$$
$$5. \ \lim_{x\to 0} { \frac {1- cos \ x}{x}} = 0$$
$$6. \ \lim_{x\to 0} { \frac {sin^{-1} \ x}{x}} = 1$$
$$7. \ \lim_{x\to 0} { \frac {tan^{-1} \ x}{x}} = 1$$
$$8. \ \lim_{x\to a} {sin^{-1} \ x} = sin^{-1} \ a, \ \vert a \vert \le 1$$
$$9. \ \lim_{x\to a} {cos^{-1} \ x} = cos^{-1} \ a, \ \vert a \vert \le 1$$
$$10.\ \lim_{x\to a} {tan^{-1} \ x} = tan^{-1} \ a, \ – \infty \lt a \lt \infty$$

### Example:

$$Find \ \lim_{x\to \infty} { \frac {sin \ x}{x}}$$
$$Let \ x = \frac 1y \ or \ y = \frac 1x$$
$$so \ that \ x \rightarrow \infty \Rightarrow y \rightarrow 0$$
$$\therefore \lim_{x\to \infty} { \frac {sin \ x}{x}} = \lim_{y\to 0} { ( y \cdot \sin \frac 1y ) }$$
$$= \lim_{y\to 0} {y} \cdot \lim_{y\to 0} { \sin \frac 1y }$$
$$= 0$$

### Limits of Log and Exponential functions:

$$\lim_{x\to 0} {e^x} = 1$$
$$\lim_{x\to 0} { \frac {e^x-1}{x}} = 1$$
$$\lim_{x\to 0} { \frac {a^x-1}{x}} = \log_{e} {a}$$
$$\lim_{x\to 0} { \frac {\log{(1+x)}}{x}} = 1$$
$$\lim_{x\to e} {\log_{e} {x}} = 1$$

### Limits of the form 1∞

$$\lim_{x\to 0} { \left( 1+ x \right) ^ \frac 1x } = e$$
$$\lim_{x\to \infty} { \left( 1+ \frac 1x \right) ^x } = e$$
$$\lim_{x\to \infty} { \left( 1+ \frac ax \right) ^x } = e^a$$

### xn Formula:

$$\lim_{x\to a} {\frac {(x^n-a^n)}{x-a}} = n(a)^{n-1}$$