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Consider
f[x] = (5x⁴ - Log[x] + e^x)/(2x⁴ + 7x + 3e^x).
Say what f[x] is doing as x -> .

In[1]:=

  Clear[f,x]; f[x_] = (25x^4-Log[x]+E^x)/(2x^4+7x+3E^x);

In[2]:=

  Plot[f[x], {x,-10,20},
  PlotStyle->
  {{Blue,Thickness[.015]}},
  AxesLabel->{"x","f[x]"}];

Answer:
The graph of the function, f(x), approaches the x-axis. Therefore the height (and value of the function) approach 0.
So, the limit of f(x) is 0.

That seems like a reasonable guess based on the graph. Let's check and see if it works. If f[x] really does go to zero as x goes to infinity then there is some x way out there for which f[x] = 0.25. See if you can find an x that will do the job.

In[3]:=

  f[x_] = (25x^4-Log[x]+E^x)/(2x^4+7x+3E^x)

Out[3]=

   x       4
  E  + 25 x  - Log[x]
  -------------------
      x            4
   3 E  + 7 x + 2 x

In[4]:=

  f[100]

Out[4]=

                100
  2500000000 + E    - Log[100]
  ----------------------------
                      100
       200000700 + 3 E

In[5]:=

  N[f[100]]

Out[5]=

  0.333333
  

In[6]:=

  N[f[1000]]

Out[6]=

  0.333333
  

In[7]:=

  N[f[10000]]

Out[7]=

  0.333333
  

Answer:
The limit is really 0.333333

You have only checked out to 10,000. How do you know the function doesn't dip below 0.333333 say out at 100,000,000,000?

Answer:
As x gets bigger and bigger the E^x is going to dominate which means the Log[x], 2x⁴, and 7x are not going to matter. That leaves us with E^x/(3 E^x) = 1/3.

That sounds really good. Can you make the graph from above look like it is giving additional evidence towards your answer?

Answer:

In[8]:=

  Plot[f[x], {x,100,1000},
  PlotStyle->
  {{Blue,Thickness[.015]}},
  AxesLabel->{"x","f[x]"}];

This graph is showing a horizontal line at a height of 0.33333 which agrees with the algebra answer.

Excellent! What was it about the previous graph that hid this fact?

Answer:
The previous graph was graphing from -10 to 20 which was too little to show global scale.