Determining Limits By Inspection

There are 2 condition to determining limits by inspection
1. x goes to positive or negative infinity
2. limit involves a polynomian devided by a polynomial
For example:
Limit (x^3 - 4) / (x^2 + x + 1) when value of x go to positive infinity
This problem on two condition:
1. polynomial over polynomial
2. x approaches infinity
The key to determining limits by inspection is in looking at power of x in the numerator and the denominator
To apply these rules:
• Must be deviding by polynomials
• X has to be approaching infinity
If the highest power of x is greater in numerator so the limit is positive or negative infinity
Example:
1. Limit (x^3 - 4) / (x^2 + x + 1) when value of x go to positive infinity

- Highest power of x in numerator is 3
- Highest power of x in denominator is 2
Since all the number are positive and x going to positive infinity so value the limit is infinity.
Limit (x^3 - 4) / (x^2 + x + 1) when value of x is going to positive infinity having positive infinity value
If you can’t tell if the answer is positive or negative infinity:
 You can substitute a large number for x
 See if you end up with a positive or negative number
 Whatever sign you get is the sign of infinity for the limit
2. Find solution!
Limit (x^3-4) / (x^4 + 3x +5) when value of x is going to positive infinity
Numerator is x^3 - 4
- Highest power of x in numerator is 3
Dominator is (x^4 + 3x +5)
- Highest power of x in denominator is 4
To know the value of function.
Deviding all of number with x which the power to divided this number is x with highest power of x from numerator and dominator. Because the highest power of x from dominator as high as numerator so we can divide all number in dominator and nominator with x^4.
Attention!
If number which divide x^4 don’t consist of variable x so the value is zero
If number which divide x^4 consist of variable x but the power variable of x in numerator more than less of x^4 so dividing this variable is zero value
If number which divide x^4 consist of x^4 so the value is a coefficient of x^4 from the numerator
Numerator = (x^3 / x^4) - (4 / x^4)
= 0 + 0 = 0
Dominator = (x^4 / x^4) + (3 / x^4) + (5 / x^4)
= 1 + 0 + 0
= 1
Limit (x^3-4) / (x^4 + 3x +5) when value of x is going to positive infinity has a zero value

3. Limit (3x^3 - 4) / (5x^3 + 3x + 5) when value of x is going to positive infinity
A LITTLE BIT TRICKER
Used when:
 Highest power of x in numerator is SAME as highest power of x in denominator
Naminator : 3x^3 - 4
Dominator : 5x^3 + 3x + 5

 Limit x going to infinity (positive or negative)
The quotient of the coefficients of two highest power
Numerator = (3x^3 / x^3) - (4 / x^3)
= 3 + 0 = 0
Dominator = (5^3 / x^3) + (3x/ x^3) + (5 / x^3)
= 5 + 0 + 0
= 5
Limit (3x^3 - 4) / (5x^3 + 3x + 5) when value of x is going to positive infinity has positive value. And the value is 3/5