what you'll learn...
Limit of Expressions evaluating to ∞∞
» Organize the sub-expressions to the following
→ a∞=0a∞=0
→ ∞±a=∞∞±a=∞
→ ∞×a=∞∞×a=∞ when a≠0a≠0
→ ∞×∞=∞
→ ∞n=∞ when n≠0
→ limx→∞xx=1
→ limx→-∞xx=1
→ limx→∞ax `=0 text( if ) 0
addition of infinity
The value of function f(x)=3x2+5x-2 at x=∞ can be found by substituting x=∞
(3x2+5x-2)
=(3(∞)2+5∞-2)
=∞
as ∞2=∞; n∞=∞; and ∞±a=∞
The value of function f(x)=13x2+5x-2 at x=∞ can be found by substituting x=∞
13x2+5x-2
=13(∞)2+5∞-2
=1∞
=0
as 1∞=0.
subtraction of infinity
The value of function f(x)=3x2-5x-2 at x=∞ is first tried with substituting x=∞
3x2-5x-2
=3∞2-5∞-2
=∞-∞
=00
as ∞2=∞; n∞=∞; and ∞±a=∞
It is noted that ∞-∞ is neither ∞ nor 0. It is indeterminate value
∞-∞
=(10)-(10)
=1-10
=00
division of infinity
The value of function f(x)=3x2+5x-2x2+x-2 at x=∞ is first tried with substituting x equals infinity.
3x2+5x-2x2+x-2
=3(∞)2+5∞-2∞2+∞-2
=∞∞
=00
as ∞2=∞; n∞=∞; and ∞±a=∞
It is noted that ∞÷∞ is neither ∞ nor 0. It is indeterminate value
∞÷∞
=(10)÷(10)
=10×01
=00
watch-out
The forms of expressions evaluate to indeterminate values when computing limit for ∞ or -∞ are ∞÷∞ and ∞-∞.
what to do
When we encounter ∞÷∞ or ∞-∞, convert the expression to one of the following forms given on left hand side
limx→∞xx=1
limx→-∞xx=1
a∞=0
∞n=∞
n∞=∞
∞±a=∞
examples
Limit of function f(x)=(3x2-5x-2) at x=∞
The function evaluates to ∞-∞ at x=∞
The limit of the function is
limx→∞(3x2-5x-2)
=limx→∞x2(2-5x-2x2)
=limx→∞x2
×limx→∞(2-5x-2x2)
=∞2×(2-0-0)
=∞
Function f(x)=3x2+5x-2x2+x-2 at x=∞
The function evaluates to ∞∞ at x=∞
The limit of the function is
limx→∞3x2+5x-2x2+x-2
=limx→∞x2(3+5x-2x2)x2(1+1x-2x2)
=limx→∞x2x2
×limx→∞3+5x-2x21+1x-2x2
=[limx→∞xx]2×3+0-01+0-0
=12×3
=3
When evaluating limits to infinity or minus infinity, simplify to known results.
Find the limit of the function limx→∞x+35x+4
The answer is '15'.
summary
Evaluating limits to ∞ or -∞: Simplify the numerical expressions to one of the following
limx→∞xx=1
limx→-∞xx=1
a∞=0
∞±a=∞
n∞=∞ where n≠0
∞×∞=∞ or
∞n=∞ where n≠0
And avoid indeterminate values ∞∞, ∞-∞, 0×∞, and ∞0 .
Outline
The outline of material to learn "limits (calculus)" is as follows.
Note : click here for detailed outline of Limits(Calculus).
→ Indeterminate and Undefined
→ Indeterminate value in Functions
→ Expected Value
→ Continuity
→ Definition by Limits
→ Geometrical Explanation for Limits
→ Limit with Numerator and Denominator
→ Limits of Ratios - Examples
→ L'hospital Rule
→ Examining a function
→ Algebra of Limits
→ Limit of a Polynomial
→ Limit of Ratio of Zeros
→ Limit of ratio of infinities
→ limit of Binomial
→ Limit of Non-algebraic Functions