How can properties be used to generate equivalent expressions?
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When we approach limits, we are finding values that are infinitesimally approaching this x-value. Essentially, we consider the approximate location that this root or limit appears. This is essential when it comes to taking Calculus, and finding the limit or rate of change of a function.
When we are attempting limits questions, there are several tests we attempt first.
1. Evaluate the limit by substituting the value of the x-value as it approaches the value (direct evaluation of a limit)
2. Rearrangement of the function, such that we can evaluate the limit.
3. (TRIGONOMETRIC PROPERTIES)
4. Using L'Hopital's Rule for indeterminate limits, such as 0/0, -infinity/infinity, or infinity/infinity.
For example:
1)
We can do this using the first and second method.
<em>Method 1: Direct evaluation:</em>
Substitute x = 0 to the function.
<em>Method 2: Rearranging the function
</em>
We can see that x - 25 can be rewritten as: (√x - 5)(√x + 5)
By rewriting it in this form, the top will cancel with the bottom easily, and our limit comes out the same.
Every example works exactly the same way, and by remembering these criteria, every limit question should come out pretty naturally.
When we are attempting limits questions, there are several tests we attempt first.
1. Evaluate the limit by substituting the value of the x-value as it approaches the value (direct evaluation of a limit)
2. Rearrangement of the function, such that we can evaluate the limit.
3. (TRIGONOMETRIC PROPERTIES)
4. Using L'Hopital's Rule for indeterminate limits, such as 0/0, -infinity/infinity, or infinity/infinity.
For example:
1)
We can do this using the first and second method.
<em>Method 1: Direct evaluation:</em>
Substitute x = 0 to the function.
<em>Method 2: Rearranging the function
</em>
We can see that x - 25 can be rewritten as: (√x - 5)(√x + 5)
By rewriting it in this form, the top will cancel with the bottom easily, and our limit comes out the same.
Every example works exactly the same way, and by remembering these criteria, every limit question should come out pretty naturally.