Find the limit. Use l'Hospital's Rule if appropriate. If there is a more elementary method, consider using it.
lim x→3 (x^2+3x−18)/(x−3)
how about I do this one, and you at least try on the others?
x^2+3x-18 = (x+6)(x-3)
So, everywhere except at x=3, f(x) = x+6
That means we expect the limit to be 3+6=9
Using the suggestion, the limit is
(2x+3)/1 = 2*6+3 = 9
To find the limit of the function (x^2+3x−18)/(x−3) as x approaches 3, we can try to directly substitute the value of x into the function. However, doing so results in an indeterminate form of 0/0.
To evaluate this limit, we can apply l'Hospital's Rule, which states that if we have an indeterminate form of the type 0/0 or ∞/∞, we can differentiate the numerator and denominator separately until we obtain a determinate form.
Let's apply l'Hospital's Rule:
Step 1: Differentiate the numerator and denominator separately:
f(x) = x^2+3x−18
g(x) = x−3
f'(x) = 2x + 3
g'(x) = 1
Step 2: Recalculate the limit using the derivatives:
lim x→3 [(2x+3)/(1)]
Substituting x=3 into the expression, we get:
(2(3)+3)/(1) = (6+3)/1 = 9/1 = 9
Therefore, the limit of the function as x approaches 3 is 9.
To find the limit of the given expression as x approaches 3, let's simplify the expression first:
The expression is (x^2 + 3x - 18)/(x - 3).
To simplify, we can factor the numerator as (x + 6)(x - 3), which gives us:
((x + 6)(x - 3))/(x - 3).
Now, we can cancel out the common factor of (x - 3) from the numerator and denominator, giving us:
(x + 6).
So the simplified expression is (x + 6).
Now, to find the limit as x approaches 3, we can substitute 3 into the simplified expression:
lim x→3 (x + 6) = 3 + 6 = 9.
Therefore, the limit of the expression as x approaches 3 is 9.
Note: In this case, we do not need to use l'Hôpital's Rule because the expression was easily simplified by factoring and canceling out the common factor. However, if the expression was not easily simplified, we could have applied l'Hôpital's Rule by taking the derivative of the numerator and denominator separately.