Limits And Continuity

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Have you ever driven a motorbike or a car?

What do we observe when we press the accelerator for any one of them?

The motorbike/car attains its maximum speed after some time int`erval say 10-15 seconds.

Now, the speed might increase from 0 km/hr initially i.e. at time 0 seconds to 100/150 km/hr after time 10-15 seconds. Now, if we are interested in finding the approximate speed of the motorbike/car at the time say 8 seconds, then we want an expected or estimated value of the speed at a particular time instant.

Here, comes the role of limits.

Similarly, if we are not able to find or determine the actual value of a function say f(x) at any given point say x = a, we try to estimate i.e. find its expected value at x = a. This is where the concept of limit comes in picture.

Limit of a Function

Let f(x) be function and x = a be any point in its domain then the limit of f(x) at x = a is denoted by $\lim_{x\to a }$ f(x) . It is the expected or estimated value of f(x) as x approaches to a.

Left Hand limit: The left hand limit of f(x) at x = a is denoted by $\lim_{x\to a^- }$ f(x). It is the expected or estimated value of f(x) at x = a when the values of f(x) near to and to the left of a are given.
Right Hand limit: The right hand limit of f(x) at x = a is denoted by $\lim_{x\to a^+}$ f(x). It is the expected or estimated value of f(x) at x= a when the values of f(x) near to and to the right of a are given
Existence of limit: If the left hand limit & right hand limit both coincide i.e. are equal then the limit exists & the common value is called the limit of the function.

If $\lim_{x\rightarrow a^-}$ f(x) = $\lim_{x\rightarrow a^+}$ f(x) = l.(say)

Then, $\lim_{x\rightarrow a}$ f(x) exists & $\lim_{x\rightarrow a}$ f(x) = l.

We read it as the limit of f(x) as x approaches to a is equal to l.

Standard Limits

Limit of polynomial function, f(x) = a₀ + a₁x + a₂x² + a₃x³ + ….+ a_nxⁿ

$\lim_{x\rightarrow a}$ f(x)= f(a)

= $\lim_{x\rightarrow a}$ (a₀ + a₁x + a₂x² + a₃x³ + …. + a_nxⁿ)

= (a₀ + a₁a + a₂a² + a₃a³ + …. + a_naⁿ)

Limit of Rational function $\lim_{x\rightarrow a}$ $\left ( \frac{f(x)}{g(x)} \right )$ = $\frac{\left ( \lim_{x\rightarrow a} f(x) \right )}{\left ( \lim_{x\rightarrow a} g(x) \right )}$ = $\frac{f(a)}{g(a)}$ provided g(x) $\neq$ 0.
$\lim_{x\rightarrow a}$ $\left ( \frac{x^n-a^n}{x-a} \right )$ =n $a^{n-1}$

Algebra of limits of functions

If f(x) and g(x) be any two functions such that $\lim_{x\rightarrow a}$ f(x) & $\lim_{x\rightarrow a}$ g(x) both exist.

$\lim_{x\rightarrow a}$ [f(x)+g(x)] = $\lim_{x\rightarrow a}$ f(x) + $\lim_{x\rightarrow a}$ g(x)
$\lim_{x\rightarrow a}$ [f(x) – g(x)] = $\lim_{x\rightarrow a}$ f(x) – $\lim_{x\rightarrow a}$ g(x)
$\lim_{x\rightarrow a}$ [f(x) g(x)] = $\lim_{x\rightarrow a}$ f(x) $\lim_{x\rightarrow a}$ g(x)
$\lim_{x\rightarrow a}$ $\left [ \frac{f(x)}{g(x)} \right ]$ = $\frac{\lim_{x\to a}f(x) }{\lim_{x\to a}g(x) }$ where g(x) $\neq$ 0 .

Examples

Now let’s consider some examples on limits and continuity.

Example 1: Find ${\lim_{x\to1}$ f(x) where f(x) = x¹¹ + 3x.

${\lim_{x\to1}$ f(x) = ${\lim_{x\to1}$ (x¹¹ + 3x) = (1)¹¹ + 3(1) = 4

Example 2: Find ${\lim_{x\to2}$ f(x) where f(x) = $\frac{x+1}{x^2+2x+3}$ .

${\lim_{x\to2}$ f(x) = ${\lim_{x\to2}$ $\left ( \frac{x+1}{x^2+2x+3} \right )$ = $\frac{\lim_{x\to2}(x+1) }{\lim_{x\to2} (x^2+2x+3)}$

= $\frac{(2+1) }{(2)^2+2(2)+3}$ = $\frac{3}{11}$

Example 3: Find ${\lim_{x\to1}$ f(x) where f(x) = $\frac{x^{10}-1}{x-1}$ .

${\lim_{x\to1}$ f(x) = ${\lim_{x\to1}$ $\left ( \frac{x^{10}-1^{10}}{x-1} \right )$ = 10 $(1)^{10-1}$ = 10 using $\lim_{x\rightarrow a}$ $\left ( \frac{x^n-a^n}{x-a} \right )$ = n $a^{n-1}$

Continuity

A function f(x) is called continuous at a point x = a in its domain if $\lim_{x\rightarrow a}$ f(x)= f(a), which can also be stated as

$\lim_{x\to a^+}$ f(x) = $\lim_{x\to a^-}$ f(x) = f(a).

Hence, if the left hand limit & right hand limit both exist, and are both equal to the value of f(x) at x = a, then the function f(x) is called continuous at a point x = a.

Example 4: Check the continuity of the function f(x) = x¹¹ + 3x at x = 1.

${\lim_{x\to1}$ f(x) = ${\lim_{x\to1}$ $(x^{11}+3x)$ = (1)¹¹ + 3(1) = 4

f(1) = (1)¹¹ + 3(1) = 4

Therefore, ${\lim_{x\to1}$ f(x) = f(1).

Hence, the function f(x) = x¹¹ + 3x is continuous at x = 1.

Example 5: Find the value of a if f(x) = 2x + a is continuous at x = 1 and f(1) = 5.

Since f(x) = 2x + a is continuous at x = 1,

∴ ${\lim_{x\to1}$ f(x)=f(1)

${\lim_{x\to1}$ (2x+a)=5

2(1)+a=5

a=5-2=3

Check Point

Find $\lim_{x\to0}$ f(x) when f(x) = x² + 2x + 3.
Find ${\lim_{x\to1}$ f(x) when f(x) = $\frac{x-1}{x^2+7x+5}$ .
Find ${\lim_{x\to2}$ f(x) when f(x) = $\frac{x^4-16}{x-2}$
Check the continuity of the function f(x) = 3x² + 5x at x = 2.
Find the value of a if f(x) = 3x² + a is continuous at x = 2 and f(2) = 17.

Answer Key

$\lim_{x\to0}$ f(x) = 3
${\lim_{x\to1}$ f(x) = 0
${\lim_{x\to2}$ f(x) = 32
f(x) = 3x² + 5x is continuous at x = 2.
a = 5

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