How do you find the limit of # 1 / (x-2)^2# as x approaches 2?

Answer 1

#lim_(xto2)1/(x-2)^2=oo#

Consider the fact that #lim_(xto0^+)1/x=oo#

also notice that

#x=2 => (x-2)^2=(2-2)^2=0^2=0#
So as #xto2# we see that #(x-2)^2 to 0#
so let #u=(x-2)^2#

then

#lim_(uto0^+)1/u=lim_(xto2)1/(x-2)^2#
and since #lim_(xto0^+)1/x=oo#
#lim_(uto0^+)1/u=oo#
and since #u=(x-2)^2# and #uto0# as #xto2#
#lim_(xto2)1/(x-2)^2=oo#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the limit of 1 / (x-2)^2 as x approaches 2, we can use algebraic manipulation.

First, we substitute 2 into the expression: 1 / (2-2)^2. This simplifies to 1 / 0^2, which is undefined.

However, we can use a different approach called factoring. By factoring the denominator, we can rewrite the expression as 1 / (x-2)(x-2).

Now, we can see that as x approaches 2, the denominator (x-2)(x-2) approaches 0.

To evaluate the limit, we can take the reciprocal of the denominator, giving us (x-2)(x-2) / 1.

Since the numerator (x-2)(x-2) approaches 0 as x approaches 2, the limit of 1 / (x-2)^2 as x approaches 2 is 0.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7