Find limit as x approaches 5 (x^2-3x-10)/(x-5)
A. 2
B. DNE
C. 0
D. 7
E. None of these
Well, let's evaluate this limit together! We can start by simplifying the expression:
(x^2 - 3x - 10)/(x - 5)
Hmm, isn't this starting to look like a bad joke? We have a denominator of (x - 5) which means we're diving by zero when x = 5. Oh no!
So, since there's a big clown party going on at x = 5, I'm sorry to inform you that the limit as x approaches 5 of this expression does not exist (DNE). It's a party that never ends!
So the answer is B. DNE.
To find the limit as x approaches 5 of the function (x^2-3x-10)/(x-5), you can either factorize or use algebraic manipulation.
Let's use algebraic manipulation:
(x^2-3x-10)/(x-5)
= [(x-5)(x+2)]/(x-5)
Now, since x-5 appears in both the numerator and denominator, we can cancel them out:
(x+2)
Now, we can substitute x=5 into the simplified expression:
(5+2)
= 7
Therefore, the limit as x approaches 5 of the function is 7.
So the correct answer is D. 7.
To find the limit as x approaches a specific value, in this case, 5, of a given function, we need to check if it is possible to directly substitute the value into the function.
Let's try substituting x = 5 into the given function:
(x^2 - 3x - 10)/(x - 5)
= ((5)^2 - 3(5) - 10)/(5 - 5)
= (25 - 15 - 10)/(0)
From the above calculation, we can see that the denominator becomes zero, which results in an undefined value. In other words, division by zero is undefined.
Since we cannot directly substitute x = 5 into the function, we need to employ an alternative method, such as factoring or canceling common terms. In this case, we can factor the numerator:
(x^2 - 3x - 10)/(x - 5)
= ((x - 5)(x + 2))/(x - 5)
Now we can cancel out the common factor of (x - 5):
(x + 2)
Therefore, the simplified function is (x + 2).
Now we can find the limit as x approaches 5 of the simplified function:
lim(x -> 5) (x + 2)
When we substitute x = 5 into the simplified function, we get:
5 + 2 = 7
Therefore, the limit as x approaches 5 of the given function is 7, represented by option D.