Growth rates of functions.
A rational function is a quotient of two polynomials. So that a rational funciton in single variable, is a quotient of two polynomials in one variable.
Let 
%2C%20q(x)#card=math&code=p%28x%29%2C%20q%28x%29&id=iGYgf) be two polynomials, of degrees 
respetively.
%3Dam%20x%5Em%20%20%2B%20a%7Bm-1%7Dx%5E%7Bm-1%7D%20%2B%20%5Cldots%20%2B%20a1%20x%20%2B%20a_0#card=math&code=p%28x%29%3Da_m%20x%5Em%20%20%2B%20a%7Bm-1%7Dx%5E%7Bm-1%7D%20%2B%20%5Cldots%20%2B%20a_1%20x%20%2B%20a_0&id=yNTZC);
%20%3D%20bn%20x%5En%20%2B%20b%7Bn-1%7D%20x%5E%7Bn-1%7D%20%2B%20%5Cldots%20%2B%20b1%20x%20%2B%20b_0#card=math&code=q%28x%29%20%3D%20b_n%20x%5En%20%2B%20b%7Bn-1%7D%20x%5E%7Bn-1%7D%20%2B%20%5Cldots%20%2B%20b_1%20x%20%2B%20b_0&id=TZfCg).
Because 
, so
%7D%7Bq(x)%7D%20%3D%20%5Cfrac%7Bam%7D%7Bb_n%7D#card=math&code=%5Clim%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bp%28x%29%7D%7Bq%28x%29%7D%20%3D%20%5Cfrac%7Ba_m%7D%7Bb_n%7D&id=vTnQ6) if 
;
if 
.
%7D%7Bq(x)%7D%20%20%3D%5Cfrac%7B0%7D%7Bbn%7D%20%3D0#card=math&code=%5Clim%7Bx%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bp%28x%29%7D%7Bq%28x%29%7D%20%20%3D%5Cfrac%7B0%7D%7Bb_n%7D%20%3D0&id=Mhbx4) if 
.
The power of the exponential function!
, 
, 
as $x \to \infty $
Exponential function grows more rapidly than power functions.
Power functions grows more rapidly than logarithmic functions.
Remark: There is a parallel theory on comparing two functions which “shrinks” more rapidly. More precisely, suppose 
and 
%3D%5Clim%7Bx%20%5Cto%20a%7D%20g(x)%3D0#card=math&code=%5Clim%7Bx%20%5Cto%20a%7Df%28x%29%3D%5Clim%7Bx%20%5Cto%20a%7D%20g%28x%29%3D0&id=dYaLM). We want to evaluate the indertminate limit %7D%7Bg(x)%7D#card=math&code=%5Clim_%7Bx%20%5Cto%20a%7D%20%5Cfrac%7Bf%28x%29%7D%7Bg%28x%29%7D&id=jtbrA). Possible answers: 
, where 
.
A good example is Example 4.41 in the Openstax Calculus textbook (vol. 1).
#card=math&code=%5Clim%7Bx%20%5Cto%200%2B%7D%20x%20%5Ccdot%20%5Cln%28x%29&id=ORMIh). If we rewrite it into the form %7D#card=math&code=%5Clim_%7Bx%20%5Cto%200%2B%7D%20%5Cfrac%7Bx%7D%7B1%2F%5Cln%28x%29%7D&id=vaq6a), it is a 
indeterminant form.
However, we do not evaluate it in this way. As a contrast, we should interpret this limit as a 
form by rewrite it as 
%2F(1%2Fx)#card=math&code=%5Clim_%7Bx%20%5Cto%200%2B%7D%20%5Cln%28x%29%2F%281%2Fx%29&id=hIWkZ).
This limit is equal to 0. This is should be helpful to understand that the power function 
“shrinks” more rapidly than logarithmic functions. Combining with the known fact that power functions grows more rapidly than lograithmic functions, we see that power functions changes more rapidly than logarithmic functions.
Aside: complexity of algorithms, the growth rate of functions play a role there.
#card=math&code=O%28%5Clog%20n%29&id=FYhfj), 
#card=math&code=O%28%5Csqrt%7Bn%7D%29&id=Dhfvx)
Roughly, %20%3D%20%5C%7B%20f(n)%20%5Cmid%20%5Clim%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bf(n)%7D%7B%5Clog%20n%7D%20%3D%20M%20%3C%20%5Cinfty%2C%20M%20%5Cneq%200%20%5C%7D#card=math&code=O%28%5Clog%20n%29%20%3D%20%5C%7B%20f%28n%29%20%5Cmid%20%5Clim%7Bn%20%5Cto%20%5Cinfty%7D%20%5Cfrac%7Bf%28n%29%7D%7B%5Clog%20n%7D%20%3D%20M%20%3C%20%5Cinfty%2C%20M%20%5Cneq%200%20%5C%7D&id=fVQcd) .
Linear approximation:
Recall this motivating problem:
Suppose the radius of our earth increases 50 cm. How much longer will the equator gain?
The function under considering shoulde be 
%3D2%5Cpi%20r#card=math&code=C%28r%29%3D2%5Cpi%20r&id=KaOEA) .Now the linearization of #card=math&code=C%28r%29&id=zUSXs) at 
is 
%2BC’(r_0)(r-r_0)#card=math&code=C%28r_0%29%2BC%27%28r_0%29%28r-r_0%29&id=UERWs) .
= the radius of our earth, approximately 
km.
The idea of differentials:
First example:
Let us pick a point 
)#card=math&code=%28a%2C%20%5Ccos%28a%29%29&id=PpYoK) on the cosine curve. The tangent line to the cosine curve at this point, is
%20%3D%20k%20(x-a)#card=math&code=y-%5Ccos%28a%29%20%3D%20k%20%28x-a%29&id=dg5z6). 
#card=math&code=k%3D-%5Csin%28a%29&id=WQ1Bb) .
Now construct a new coordinating system at )#card=math&code=%28a%2C%20%5Ccos%28a%29%29&id=K1PDP) with this point the new origin. The equation to the tangent line inside this new coordinating system will be simple: 
%5Cvert%7Bx%3Da%7D%20X#card=math&code=Y%3D%20%5Cfrac%7Bd%20%7D%7Bdx%7D%20%5Ccos%28x%29%5Cvert%7Bx%3Da%7D%20X&id=J5lOY).
Same scenario can happen at each point 
)#card=math&code=%28b%2C%20%5Ccos%28b%29%29&id=XMSj4) on the cosine curve.
Second example:
Now consider another function: 
%3Dx%5E2%2B2x#card=math&code=y%3Df%28x%29%3Dx%5E2%2B2x&id=OhdSg). At the point 
#card=math&code=%28-3%2C%203%29&id=EesvY), same story can be told as the construction for cosinde curve above. The tangent line in the new coordinating system, is
, 
%5Cvert%7Bx%3D-3%7D%3D-4#card=math&code=k%20%3D%20%5Cfrac%7Bd%7D%7Bdx%7D%20%28x%5E2%2B2x%29%5Cvert%7Bx%3D-3%7D%3D-4&id=jCWzn).
The slope 
, which now is the ratio 
, is always equal to 
.
When we are considering the problem at 
, then 
.
