lim x->5 (6-x)/(x-5)^2
To evaluate the limit of (6-x)/(x-5)^2 as x approaches 5, we can try direct substitution to see if it is indeterminate.
Plugging in x = 5 into the expression, we get (6-5)/(5-5)^2 = 1/0^2 = 1/0.
Since we obtained the indeterminate form 1/0, we cannot evaluate the limit by direct substitution.
To simplify the expression and evaluate the limit, we can factor the numerator as:
(6-x)/(x-5)^2 = -1/(x-5)
Now, we can plug in x = 5 into this simplified expression:
-1/(5-5) = -1/0.
Again, we obtained the indeterminate form -1/0.
To further simplify the expression and evaluate the limit, we can use the limit properties.
We can rewrite the expression as:
(6-x)/(x-5)^2 = -(1/(x-5))
As x approaches 5 from both sides, the expression -(1/(x-5)) approaches negative infinity, since the denominator (x-5) approaches 0 from the right side, while the numerator -1 remains constant.
Therefore, the limit of (6-x)/(x-5)^2 as x approaches 5 is negative infinity: lim x->5 (6-x)/(x-5)^2 = -∞.
the answer is positive infinity
Apologies for the mistake in my previous response. You are correct.
Let's reassess the limit:
We have the expression (6-x)/(x-5)^2. We can begin by simplifying the expression:
(6-x)/(x-5)^2 = -1/(x-5)
Now, let's evaluate the limit as x approaches 5:
lim x->5 -1/(x-5)
As x approaches 5, we have:
lim x->5- -1/(x-5) = -1/0- = -∞ (approaching negative infinity from the left side)
lim x->5+ -1/(x-5) = -1/0+ = -∞ (approaching negative infinity from the right side)
Since the limit approaches negative infinity from both sides, the correct answer is lim x->5 (6-x)/(x-5)^2 = -∞.
To find the limit of the given function as x approaches 5, follow these steps:
Step 1: Start by simplifying the function.
The function (6-x)/(x-5)^2 can be simplified by factoring out the numerator.
(6-x)/(x-5)^2 = -(x-6)/(x-5)^2
Step 2: Identify any potential issues.
At first glance, it appears that this function has a removable discontinuity at x=5 since (x-5)^2 is in the denominator.
Step 3: Apply limit properties for rational functions.
Since the denominator contains a factor of (x-5)^2, we can apply the limit properties for rational functions. Specifically, if f(x)/g(x) is a fraction where f(x) and g(x) are continuous functions and g(x) ≠ 0 for all x, then:
lim x->a f(x)/g(x) = (a) provided that lim x->a g(x) ≠ 0.
Step 4: Evaluate the limit.
Using the limit properties, we can ignore the fact that (x-5)^2 becomes zero at x=5 and evaluate the limit directly:
lim x->5 -(x-6)/(x-5)^2
Plugging in x=5 directly would result in an undefined expression (0/0). Therefore, we need to simplify further.
Step 5: Simplify the function.
To eliminate the (x-5)^2 term, we can factor out (x-5)^2 from the numerator:
lim x->5 -(x-6)/(x-5)^2 = lim x->5 -(1/(x-5))
Now we can evaluate the limit by substituting x=5:
lim x->5 -(1/(x-5)) = -1/(5-5) = -1/0
Since we get an undefined expression of -1/0, this suggests that the limit does not exist.
Therefore, the limit of (6-x)/(x-5)^2 as x approaches 5 is undefined.
To evaluate the limit of the expression (6-x)/(x-5)^2 as x approaches 5, we can use algebraic manipulation and the properties of limits.
Let's start by simplifying the expression using algebraic manipulation:
(6-x)/(x-5)^2 = -1/(x-5)
Now, let's analyze the expression -1/(x-5) as x approaches 5.
When x approaches 5, the denominator (x-5) approaches 0, which results in a division by zero. This situation is called an indeterminate form, and we cannot directly substitute the value of x into the expression.
To evaluate this type of limit, we can use a method called "rationalizing the numerator." This involves multiplying the numerator and denominator by a suitable conjugate to eliminate the division by zero.
In this case, multiplying both the numerator and denominator by (x-5), we get:
(-1/(x-5)) * ((x-5)/(x-5)) = -1
So, as x approaches 5, the expression (6-x)/(x-5)^2 approaches -1.
Therefore, the limit of (6-x)/(x-5)^2 as x approaches 5 is -1.