Question

What is the limit of (x−1)(x−2)(x−3)(x−4)x−5(5x−1)5 as x goes to infinity?

Answer 1

You can separate this into multiple limits using their multiplicative properties.

=

lim

x

→

∞

x

−

1

5

x

−

1

⋅

lim

x

→

∞

x

−

2

5

x

−

1

⋅

lim

x

→

∞

x

−

3

5

x

−

1

⋅

lim

x

→

∞

x

−

4

5

x

−

1

⋅

lim

x

→

∞

x

−

5

5

x

−

1

Then, we can "simplify" this down using the fact that as

x

becomes really large, the evaluation

x

5

x

≈

1

5

on each limit becomes more and more accurate (with

x

>>

1

, the

1

becomes insignificant at large

x

).

Since there are five of these limits, you get

(

1

5

)

⋅

(

1

5

)

⋅

(

1

5

)

⋅

(

1

5

)

⋅

(

1

5

)

=

1

5

5

, or

1

3125

.