Finding limits of functions
C, 402 bytes

Plenty of room for golfing, just proof of concept to begin with :P

Compiles under clang, osx.

Note: For linux you may have to change the code to use something like: "ld -shared f.o -o f.so -lm"

#include&lt;dlfcn.h&gt;
#include&lt;stdio.h&gt;
float e,r;
main(int i,char**v){
    // create and compile temporary c file for 'f(x)'
    // note: include offset within function
    FILE*s=fopen("f.c","w");
    fprintf(s,"#include&lt;math.h&gt;\nfloat f(float x){x+=%s;return %s;}",v[2],v[1]); 
    fclose(s);
    system("cc -fPIC -c f.c;ld -bundle f.o -o f.so -lm");
    float(*f)(float)=dlsym(dlopen("f.so",0),"f");

    for(i=0;1;e=(e==0)?1e-37:e*2){
        r=f(e);
        if(isinf(r))i=1;else if(!isnan(r)){
            i?printf("%cINF\n",r&gt;0?43:45):printf("%f\n",r);
            break;
        }
    }
}


Run as:

./a.out "sin(x)/x" 0
./a.out "1/(x*x)" 0
./a.out "(x*x-x)/(x-1)" 1
./a.out "pow(2,x)" 3


Results:

1.000000
+INF
1.000000
8.000000