Determining Limits Algebraically
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Key Questions

When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
#lim_{x to 0^}1/x=1/{0^}=infty# 1 is divided by a number approaching 0, so the magnitude of the quotient gets larger and larger, which can be represented by
#infty# . When a positive number is divided by a negative number, the resulting number must be negative. Hence, then limit above is#infty# .(Caution: When you have infinite limits, those limts do not exist.)
Here is another similar example.
#lim_{x to 3^+}{2x+1}/{x+3}={2(3)+1}/{(3^+)+3}={5}/{0^+}=infty# If no quantity is approaching zero, then you can just evaluate like a twosided limit.
#lim_{x to 1^}{12x}/{(x+1)^2}={12(1)}/{(1+1)^2}=1/4# I hope that this was helpful.

By eliminating common factors, we can find
#lim_{a to 0}{(a3)^29}/a=6# .Let us look at some details.
First, we notice that both the numerator and the denominator approach
#0# as#a# approaches#0# , which indicates that they share a common factor#a# .#lim_{a to 0}{(a3)^29}/a# by multiplying out the square,
#=lim_{a to 0}{a^26a+99}/a# by cancelling out the 9's,
#=lim_{a to 0}{a^26a}/a# by factoring out
#a# ,#=lim_{a to 0}{a(a6)}/a# by cancelling
#a# 's,#=lim_{a to 0}(a6)# by plugging in
#a=0# ,#=06=6# 
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Limits

1Introduction to Limits

2Determining One Sided Limits

3Determining When a Limit does not Exist

4Determining Limits Algebraically

5Infinite Limits and Vertical Asymptotes

6Limits at Infinity and Horizontal Asymptotes

7Definition of Continuity at a Point

8Classifying Topics of Discontinuity (removable vs. nonremovable)

9Determining Limits Graphically

10Formal Definition of a Limit at a Point

11Continuous Functions

12Intemediate Value Theorem

13Limits for The Squeeze Theorem