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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsLimit of a functionLimit of Algebraic Expressions

### Limit of functions evaluating to oo

Finding limit of standard ratios evaluating to oo/oo or oo-oo is explained with examples.

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What is the value of function f(x)=3x^2+5x-2 at x=oo?

• oo
• oo
• -2
• 0

The answer is 'oo'

By substitution x=oo
(3x^2+5x-2)
quad quad = (3(oo)^2+5 oo - 2)
quad quad = oo
as oo^2=oo; n oo = oo; and oo +- a = oo

What is the value of function f(x)=1/(3x^2+5x-2) at x=oo?

• oo
• -2
• 0
• 0

The answer is '0'

By substitution x=oo
1/(3x^2+5x-2)
quad quad = 1/(3(oo)^2+5 oo - 2)
quad quad = 1/oo
quad quad = 0
as 1/infinity = 0.

What is the value of function f(x)=3x^2-5x-2 at x=oo?

• oo-oo
• oo-oo
• 1
• 0

The answer is 'oo - oo'

By substitution x=oo
3x^2-5x-2
quad quad = 3 oo^2 - 5 oo -2
quad quad = oo - oo
quad quad = 0/0

as oo^2=oo; n oo = oo; and oo +- a = oo

Note: The limit to this function is explained after few pages.

What is the value of oo - oo?

• oo
• 1
• 0
• indeterminate value
• indeterminate value

The answer is 'indeterminate value'

The equivalence can be explained with
oo - oo
quad quad = (1/0) - (1/0)
quad quad = (1-1)/0
quad quad = 0/0

What is the value of function f(x)=color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2) at x=oo?

• oo/oo
• oo/oo
• 1

The answer is 'oo/oo'

By substitution x=oo
color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2)
quad quad = color(deepskyblue)(3(oo)^2+5 oo - 2)/color(coral)(oo^2+oo-2)
quad quad = color(deepskyblue)(oo)/color(coral)(oo)
quad quad = 0/0

as oo^2=oo; n oo = oo; and oo +- a = oo

Note: The limit to this function is explained after few pages.

What is the value of oo/oo?

• oo
• 1
• 0
• indeterminate value
• indeterminate value

The answer is 'indeterminate value'

The equivalence can be explained with
oo/oo
quad quad = (1/0) -:(1/0)
quad quad = 1/0 xx 0/1
quad quad = 0/0

What forms of expressions evaluate to indeterminate values when computing limit for oo or -oo?

• oo xx oo and oo + oo
• oo -: oo and oo - oo
• oo -: oo and oo - oo

The answer is 'oo -: oo and oo - oo'

When we encounter oo -: oo or oo - oo, convert the expression to one of the following forms given on left hand side
lim_(x->oo) x/x = 1
lim_(x->-oo) x/x = 1
a/oo = 0
oo^n=oo
n oo = oo
oo +- a = oo

Limit of function f(x)=(3x^2-5x-2) at x=oo
The function evaluates to oo-oo at x=oo

The limit of the function is
lim_(x->oo) (3x^2-5x-2)
quad quad = lim_(x->oo) x^2(2-5/x - 2/x^2)
quad quad = lim_(x->oo) x^2
quad quad quad quad xx lim_(x->oo) (2-5/x - 2/x^2)
quad quad = oo^2 xx (2-0-0)
quad quad = oo

Function f(x)=color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2) at x=oo
The function evaluates to oo/oo at x=oo

The limit of the function is
lim_(x->oo) color(deepskyblue)(3x^2+5x-2)/color(coral)(x^2+x-2)
quad quad = lim_(x->oo) color(deepskyblue)(x^2(3+5/x-2/x^2))/color(coral)(x^2(1+1/x-2/x^2))
quad quad = lim_(x->oo) color(deepskyblue)(x^2)/color(coral)(x^2)
quad quad quad quad xx lim_(x->oo)color(deepskyblue)(3+5/x-2/x^2)/color(coral)(1+1/x-2/x^2)
quad quad = [lim_(x->oo) color(deepskyblue)(x)/color(coral)(x)]^2 xx color(deepskyblue)(3+0-0)/color(coral)(1+0-0)
quad quad = 1^2 xx 3
quad quad = 3

When evaluating limits to infinity or minus infinity, simplify to known results.

Evaluating limits to oo or -oo: Simplify the numerical expressions to one of the following
lim_(x->oo) x/x = 1
lim_(x->-oo) x/x = 1
a/oo = 0
oo +- a = oo
n oo = oo where n!=0
oo xx oo = oo or
oo^n=oo where n!=0
And avoid indeterminate values oo/oo, oo-oo, 0 xx oo, and oo^0 .

Solved Exercise Problem:

Find the limit of the function lim_(x->oo) (x+3)/(5x+4)

• 1/5
• 1/5
• 5
• oo
• 0

The answer is '1/5'

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