How to Find the Limit of (1 - x)/[(3 - x)^2] as x Approaches 3?

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SUMMARY

The limit of the function (1 - x)/[(3 - x)^2] as x approaches 3 is definitively negative infinity. This conclusion is reached by analyzing the behavior of the denominator, which approaches zero, while the numerator remains negative. Graphical analysis supports this finding, but algebraic methods can also confirm the limit by recognizing the implications of dividing by a very small number. The discussion emphasizes the importance of understanding limits in calculus without solely relying on graphical representations.

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  • Understanding of limits in calculus
  • Familiarity with algebraic manipulation of functions
  • Knowledge of graphing functions and interpreting their behavior
  • Experience with evaluating limits using tables of values
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  • Learn how to apply L'Hôpital's Rule for indeterminate forms
  • Explore the use of tables of values for limit evaluation
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Find the limit of (1 - x)/[(3 - x)^2] as x---> 3.

I could not find the limit using algebra. So, I decided to graph the given function.
I can see from the graph on paper that the limit is negative infinity.
How is this done without graphing?
 
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Problem 1.5.35.
Odd numbered.
Look up the answer.
 
I don't understand why you can't find the answer using algebra?

If you divide by a big number, your total amount is small.

If you divide by a small number, your total amount is big.

Here, the denominator is getting closer to 0 (so very small), so what do you think happens to the whole amount?
 
Prove It said:
I don't understand why you can't find the answer using algebra?

If you divide by a big number, your total amount is small.

If you divide by a small number, your total amount is big.

Here, the denominator is getting closer to 0 (so very small), so what do you think happens to the whole amount?

I am learning this material on my own with very limited time on my hand. Be a little more understanding in your reply. The limit is negative infinity. How is this done using a table of values?
 
nycmathdad said:
I am learning this material on my own with very limited time on my hand. Be a little more understanding in your reply. The limit is negative infinity. How is this done using a table of values?

Why do you need a table of values at all? You have already established that the limit is $-\infty$ because the denominator gets extremely small, and the function has negative values on both sides.
 
Prove It said:
Why do you need a table of values at all? You have already established that the limit is $-\infty$ because the denominator gets extremely small, and the function has negative values on both sides.

Ok. Take it easy, bro. Happy Resurrection Sunday. Relax.
 

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