How do you find the limit #lim_(x->2)(x^2+x-6)/(x-2)# ?

Answer 1

Start by factoring the numerator:

#= lim_(x->2) (((x + 3)(x-2))/(x-2))#

We can see that the #(x - 2)# term will cancel off. Therefore, this limit is equivalent to:

#= lim_(x->2) (x + 3)#

It should now be easy to see what the limit evaluates to:

#= 5#

Let's take a look at a graph of what this function would look like, to see if our answer agrees:

The "hole" at #x = 2# is due to the #(x - 2)# term in the denominator. When #x = 2#, this term becomes #0#, and a division by zero occurs, resulting in the function being undefined at #x = 2#. However, the function is well-defined everywhere else, even when it gets extremely close to #x = 2#.

And, when #x# gets extremely close to #2#, #y# gets extremely close to #5#. This verifies what we demonstrated algebraically.

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Answer 2

To find the limit of the given expression, we can directly substitute the value of x into the expression. However, since the expression is undefined when x = 2, we need to use algebraic manipulation to simplify it. By factoring the numerator, we get (x-2)(x+3). Canceling out the common factor of (x-2) in the numerator and denominator, we are left with the limit as x approaches 2 of (x+3). Substituting x = 2 into this expression, we find that the limit is equal to 5.

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Answer from HIX Tutor

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.

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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