Can someone explain the "LIATE" rule (1 Viewer)

[ohlookmonkeys]

I heard you can use it to determine which functions increases faster as they approach limits.
I'm using this for the addition of graphs exercise btw, haven't learnt differentiation of inverse trig and logs.

 

[Carrotsticks]

I heard you can use it to determine which functions increases faster as they approach limits.
I'm using this for the addition of graphs exercise btw, haven't learnt differentiation of inverse trig and logs.

LIATE is an easy way to explain how 'powerful' functions are and which functions 'dominate' others.

The faster a function converges to infinity (or more technically 'diverges'), the more 'powerful' it is.

Here are some examples:

L = ln(x)

I = arccos(x)

A = x^n

T = cos (x)

E = e^x

it is arranged according to increasing power, where e^x is the most 'powerful' curve and the ln(x) is the least 'powerful' curve.

For example, suppose I had the function:



Suppose I want to find the limit as x approaches infinity.

The bottom exponential function e^x is more 'powerful' than the normal algebraic function x, so imagine that it 'pulls' the curve down.

Sorry if this is a bit ambiguous, the process of dominating functions is more easily explained in person, rather than on the internet in text form.

 

[SpiralFlex]

That's funny, I saw/heard someone teach this rule of thumb at the library today. O_O

 

[ohlookmonkeys]

L = ln(x)

I = arccos(x)

A = x^n

T = cos (x)

E = e^x

it is arranged according to increasing power, where e^x is the most 'powerful' curve and the ln(x) is the least 'powerful' curve.

For "I" and "T", do they include all other trigonometric functions? Because tangent seems to be more "powerful" than cosine and sine.

 

[SpiralFlex]

For "I" and "T", do they include all other trigonometric functions? Because tangent seems to be more "powerful" than cosine and sine.

Inverse trigonometric functions (I) and your standard trigonometric functions (T)

 

[Carrotsticks]

For "I" and "T", do they include all other trigonometric functions? Because tangent seems to be more "powerful" than cosine and sine.

Remember that LIATE is a guide, not a rule. It does not always work.

For example, according to LIATE, Trigonometric functions should be more 'powerful' than algebraic functions ie: sin(x) > x

However, observe the interval . The line y=x is clearly dominant over the curve y=sin(x). Furthermore, the tan(x) curve is more dominant over the y=x line within the same interval.

The fact that y=x dominates y=sin(x) can be further verified by observing the limit as x --> infinity of the curve y=sin(x)/x (which oscillates, but slowly converges to 0).

When comparing T with T, or I with I etc, you obviously cannot use LIATE and you may have to resort to a geometrical argument.

You are correct in saying that the tangent function is more 'powerful' than the cosine and sine function. If you had a question involving tan and sin/cos, then the tan function would be the dominating one.

The key thing to remember is what I said earlier...that LIATE is a guide, not a rule.

 

[ohlookmonkeys]

Remember that LIATE is a guide, not a rule. It does not always work.

For example, according to LIATE, Trigonometric functions should be more 'powerful' than algebraic functions ie: sin(x) > x

However, observe the interval . The line y=x is clearly dominant over the curve y=sin(x). Furthermore, the tan(x) curve is more dominant over the y=x line within the same interval.

The fact that y=x dominates y=sin(x) can be further verified by observing the limit as x --> infinity of the curve y=sin(x)/x (which oscillates, but slowly converges to 0).

When comparing T with T, or I with I etc, you obviously cannot use LIATE and you may have to resort to a geometrical argument.

You are correct in saying that the tangent function is more 'powerful' than the cosine and sine function. If you had a question involving tan and sin/cos, then the tan function would be the dominating one.

The key thing to remember is what I said earlier...that LIATE is a guide, not a rule.

thanks, u helped me a lot

 

[cutemouse]

LIATE also fails with

 

[Carrotsticks]

LIATE also fails with

It fails if you blindly let:



However, it does work if you let:



Because of course, this actually allows for the to get integrated.

 

[cutemouse]

However, it does work if you let:

Yes but that isn't the LIATE rule...

 

[Carrotsticks]

Yes but that isn't the LIATE rule...

When I use LIATE as a guide, I usually try my best to make the exponential function integrate-able (is that a word? lol). Even if this includes combining it with another function.

I understand that throwing the in front of it may sort of 'break' that rule, but the primary function being integrated is the exponential function, not the .

The is just there to facilitate the integration process.

 

[AAEldar]

When I use LIATE as a guide, I usually try my best to make the exponential function integrate-able (is that a word? lol). Even if this includes combining it with another function.

I understand that throwing the in front of it may sort of 'break' that rule, but the primary function being integrated is the exponential function, not the .

The is just there to facilitate the integration process.

Integrable would be the word you're looking for

 

[Carrotsticks]

Integrable would be the word you're looking for

That's what I originally thought, so I tried that, then I got the red squiggly underline.

Then again, spell check isn't always the most reliable of sources when it comes to vocabulary =/

Thanks for verifying!

 

[AAEldar]

That's what I originally thought, so I tried that, then I got the red squiggly underline.

Then again, spell check isn't always the most reliable of sources when it comes to vocabulary =/

Thanks for verifying!

Hahaha yea even saying it out loud (at least to me) doesn't sound completely correct but it is!

I prefer to say something like "the function we're trying to integrate" instead of "integrable function" though.