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Summary Rules and Techniques for Solving Limits

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In these notes, you will learn step by step how to solve each type of mathematical limit (determination), where you will learn each type of determination, approaches on how to solve them, and solved exercises detailing every single step. If you are looking for a straight-forward guide on limits, this is exactly what you are looking for, no-nonsense is present in these notes, and they go straight to the point, in the exact way I wished I was taught when I had to learn them at HS, I made these notes myself after making my own research on the best and easiest methods, and it got me though in no time, now, I study physics and mathematics at college, and once again, this is going to save you a lot of time if you are looking towards excelling in calculus.

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Limits
Anibal Garcia
May 8, 2020



Contents
1 Indeterminate Form k0 , k 6= 0 2
0
2 Indeterminate Form 0
4
∞
3 Indeterminate Form ∞
7

4 Indeterminate Form ∞ − ∞ 12

5 Indeterminate Form 0 · ∞ 16




1

,1 Indeterminate Form k0 , k 6= 0
ˆ If we get an indeterminate form of k0 after calculating the limit of a function, we
will need to calculate the lateral limits.


ˆ If a the function has a limit, both laterals will have the same value and therefore
the limit will exist. If the values do not match, the limit for the function will not
exist.

Example 1:
3x + 7
lim
x→0 2x4


After substituting x for 0, we verify that:
3x + 7 7
lim 4
−→
x→0 2x 0
Therefore, we find the lateral limits:
3x + 7
lim+ = +∞
x→0 2x4


3x + 7
lim− = +∞
x→0 2x4


Both lateral limits have the same value ∴
3x + 7 7
∃ lim 4
→ = +∞
x→0 2x 0




2

, Example 2:

We calculate the limit for the function:
6x + 2
lim1
x→ 2 4x − 2
1
After substituting x for 2
we verify that:

6x + 2 5
lim1 −→
x→ 2 4x − 2 0

We then find the values for the lateral limits:
6x + 2 5
lim+ = + = +∞
x→ 12 4x − 2 0


6x + 2 5
lim− = − = −∞
x→ 12 4x − 2 0
The values for the lateral limits don’t match:
6x + 2
∴ @ lim1
x→ 2 4x − 2




3

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Browse related subjects

Subjects

  • limits
  • limits calculus
  • calculus
  • indeterminations
  • rules for limits
  • solving limits
  • calculus 1
  • mathematics
  • precalculus

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