Find the Limit

Lim x-5/ (1/x)-(1/5)
x->5

can I assume that 1/x is going to infinity so (1/x) is zero? so then I will be left with:

5-5/ (-1/5)

0/ (-1/5)
=limit is 0?

no, x ---> 5, so 1/x ----> 1/5

you would get 0/0
When that happens, you are guaranteed that your expression factors somehow.

how about multiplying top and bottom by 5x.
Then

lim(5x^2 - 25x)/(5-x) as x ---> 5
= lim 5x(x-5)/(5-x) as x ---> 5
= lim 5x(-1) as x ---> 5
= -25

ahhh i see.

dang, then I got this one wrong on the test.

thank you!

To find the limit of a function as x approaches a certain value, we need to evaluate the function when x is very close to that value. Let's calculate the limit step by step:

First, we have the expression lim(x -> 5) [(x - 5) / (1/x - 1/5)].

We cannot assume that (1/x) goes to infinity as x approaches 5. In this case, we need to simplify the expression before we evaluate the limit.

To do this, first, find a common denominator for 1/x and 1/5, which is the least common multiple (LCM) of x and 5. The LCM of x and 5 is simply 5x since x and 5 do not share any common factors other than 1.

Next, simplify the expression. The numerator, (x - 5), does not require any further simplification. For the denominator, we need to multiply 1/x by 5/5 and multiply 1/5 by x/x:

[(x - 5) / (1/x - 1/5)] = [(x - 5) / (5/x - x/5)]

Now, we can simplify further. To combine the denominators, we can take the product of the two denominators:

[(x - 5) / (5/x - x/5)] = [(x - 5) / ((5x - x^2)/5x)]

Next, invert and multiply by the reciprocal of the denominator:

[(x - 5) / ((5x - x^2)/5x)] = [(x - 5) * (5x/ (5x - x^2))]

Simplify the expression:

[(x - 5) * (5x/ (5x - x^2))] = [(5x - 25)/(5x - x^2)]

Now, we can evaluate the limit as x approaches 5:

lim(x -> 5) [(5x - 25) / (5x - x^2)]

Plugging in x = 5, we get:

[(5*5 - 25) / (5*5 - 5^2)] = 0/0

The expression evaluates to 0/0, which is an indeterminate form. Therefore, we cannot determine the limit by direct substitution.

To find the limit in this case, we can use algebraic techniques such as factoring or canceling common factors.