# How would you graph #y = ln(x^2) # without a calculator?

I tried with a....simplification...!

I would try using the base of the natural log (which is the irrational number

So basically I would graph:

Where the "real" graph is:

graph{ln(x^2) [-10, 10, -5, 5]}

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To graph ( y = \ln(x^2) ) without a calculator:

- Understand the function: ( \ln(x^2) ) means taking the natural logarithm of the square of ( x ).
- Identify key points:
- Since ( \ln(x^2) ) is undefined for ( x \leq 0 ), we will focus on ( x > 0 ).
- When ( x = 1 ), ( \ln(1^2) = \ln(1) = 0 ).
- As ( x ) increases, ( \ln(x^2) ) also increases, but it increases at a decreasing rate due to the logarithmic nature.
- The function is symmetrical around the y-axis.

- Plot key points:
- Plot the point (1, 0) since ( \ln(1^2) = 0 ).
- Plot other points symmetrically around the y-axis.

- Draw the curve:
- Connect the plotted points smoothly, noting the asymptotic behavior as ( x ) approaches 0 from the right. The curve should approach negative infinity.
- Label the axes and any key points if necessary.

- Verify the graph:
- Ensure the graph reflects the properties of the function, such as being continuous and never crossing the x-axis.

By following these steps, you can graph ( y = \ln(x^2) ) without using a calculator.

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When evaluating a one-sided 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.

When evaluating a one-sided 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.

When evaluating a one-sided 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.

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