Calculating Limit at x=a: Not Equivalent?

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In summary, when using direct substitution to calculate the limit at x = a, some functions may be simplified which can change the behavior of the function at x = a. In the example provided, the original function f(x) is not defined at x = 1, while the simplified function is defined. This is because the simplification assumes a certain operation, which is not valid at x = 1. Therefore, not all functions can be simplified without affecting their defined points.
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Shaybay92
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When using direct substitution to calculate the limit at x = a some functions are simplified so that x = a is actually defined. For example:

Lim x->1 [(x^2 - 1)/(x-1)]

Limx->1 [(x-1)(x+1)/(x-1)]

Lim x->1 [(x+1)] = 2 (when x=1 is substituted in)

I understand that they can have the same limits despite not both being defined at x=1, however, what I don't get is why the original f(x) isn't defined, but the second one is. How can two equivalent functions not be defined at the same points? Can't all functions be simplified by factorising without jeapordizing where they are actually defined? This makes no sense to me...
 
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When you made that simplification, you assumed that [tex]\frac{x-1}{x-1}=1[/tex]. That's only true when [tex]x\ne1[/tex]. When x=1, it's not a valid operation, which is why the behavior of the function is changed there.
 
  • #3
[tex]\frac{x^2- 1}{x- 1}[/tex] is not defined at x= 1 because [tex]\frac{0}{0}[/tex] is not defined.
 

FAQ: Calculating Limit at x=a: Not Equivalent?

What is the definition of a limit at x=a?

The limit at x=a is a mathematical concept that describes the behavior of a function as the input value (x) approaches a specific point (a). It is denoted by lim f(x) as x approaches a, and it represents the value that the function approaches as x gets closer and closer to a.

How is a limit at x=a calculated?

The limit at x=a is calculated by evaluating the function at x values that are increasingly closer to the given value of a. This process is called taking the limit, and it involves using algebraic techniques or graphical methods to determine the value that the function approaches as x approaches a.

What are the conditions for a limit at x=a to exist?

For a limit at x=a to exist, the function must approach a finite value as x approaches a from both the left and right sides. This means that the left and right-hand limits must be equal at x=a. Additionally, the function must be defined at x=a, but it does not need to be continuous.

Can a limit at x=a be undefined?

Yes, a limit at x=a can be undefined. This can occur if the left and right-hand limits are not equal, or if one or both of the limits approach infinity. It can also happen if the function is not defined at x=a.

How is calculating a limit at x=a different from finding a derivative?

While both involve evaluating a function at a specific point, calculating a limit at x=a is focused on determining the value that the function approaches as x gets closer to a. Finding a derivative, on the other hand, involves determining the instantaneous rate of change of a function at a specific point. In other words, a limit at x=a is concerned with the behavior of a function as x approaches a, while a derivative is concerned with the behavior of a function at a specific point.

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