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Find the following limit:
lim_(x->10) (x^27-10^27)/(x^26-10^26)

 

(x^27-10^27)/(x^26-10^26) = (x^26+10 x^25+100 x^24+1000 x^23+10000 x^22+100000 x^21+1000000 x^20+10000000 x^19+100000000 x^18+1000000000 x^17+10000000000 x^16+100000000000 x^15+1000000000000 x^14+10000000000000 x^13+100000000000000 x^12+1000000000000000 x^11+10000000000000000 x^10+100000000000000000 x^9+1000000000000000000 x^8+10000000000000000000 x^7+100000000000000000000 x^6+1000000000000000000000 x^5+10000000000000000000000 x^4+100000000000000000000000 x^3+1000000000000000000000000 x^2+10000000000000000000000000 x+100000000000000000000000000)/(x^25+10 x^24+100 x^23+1000 x^22+10000 x^21+100000 x^20+1000000 x^19+10000000 x^18+100000000 x^17+1000000000 x^16+10000000000 x^15+100000000000 x^14+1000000000000 x^13+10000000000000 x^12+100000000000000 x^11+1000000000000000 x^10+10000000000000000 x^9+100000000000000000 x^8+1000000000000000000 x^7+10000000000000000000 x^6+100000000000000000000 x^5+1000000000000000000000 x^4+10000000000000000000000 x^3+100000000000000000000000 x^2+1000000000000000000000000 x+10000000000000000000000000):
lim_(x->10) (x^26+10 x^25+100 x^24+1000 x^23+10000 x^22+100000 x^21+1000000 x^20+10000000 x^19+100000000 x^18+1000000000 x^17+10000000000 x^16+100000000000 x^15+1000000000000 x^14+10000000000000 x^13+100000000000000 x^12+1000000000000000 x^11+10000000000000000 x^10+100000000000000000 x^9+1000000000000000000 x^8+10000000000000000000 x^7+100000000000000000000 x^6+1000000000000000000000 x^5+10000000000000000000000 x^4+100000000000000000000000 x^3+1000000000000000000000000 x^2+10000000000000000000000000 x+100000000000000000000000000)/(x^25+10 x^24+100 x^23+1000 x^22+10000 x^21+100000 x^20+1000000 x^19+10000000 x^18+100000000 x^17+1000000000 x^16+10000000000 x^15+100000000000 x^14+1000000000000 x^13+10000000000000 x^12+100000000000000 x^11+1000000000000000 x^10+10000000000000000 x^9+100000000000000000 x^8+1000000000000000000 x^7+10000000000000000000 x^6+100000000000000000000 x^5+1000000000000000000000 x^4+10000000000000000000000 x^3+100000000000000000000000 x^2+1000000000000000000000000 x+10000000000000000000000000)

 

lim_(x->10) (x^26+10 x^25+100 x^24+1000 x^23+10000 x^22+100000 x^21+1000000 x^20+10000000 x^19+100000000 x^18+1000000000 x^17+10000000000 x^16+100000000000 x^15+1000000000000 x^14+10000000000000 x^13+100000000000000 x^12+1000000000000000 x^11+10000000000000000 x^10+100000000000000000 x^9+1000000000000000000 x^8+10000000000000000000 x^7+100000000000000000000 x^6+1000000000000000000000 x^5+10000000000000000000000 x^4+100000000000000000000000 x^3+1000000000000000000000000 x^2+10000000000000000000000000 x+100000000000000000000000000)/(x^25+10 x^24+100 x^23+1000 x^22+10000 x^21+100000 x^20+1000000 x^19+10000000 x^18+100000000 x^17+1000000000 x^16+10000000000 x^15+100000000000 x^14+1000000000000 x^13+10000000000000 x^12+100000000000000 x^11+1000000000000000 x^10+10000000000000000 x^9+100000000000000000 x^8+1000000000000000000 x^7+10000000000000000000 x^6+100000000000000000000 x^5+1000000000000000000000 x^4+10000000000000000000000 x^3+100000000000000000000000 x^2+1000000000000000000000000 x+10000000000000000000000000)  =  (100000000000000000000000000+10000000000000000000000000 10+1000000000000000000000000 10^2+100000000000000000000000 10^3+10000000000000000000000 10^4+1000000000000000000000 10^5+100000000000000000000 10^6+10000000000000000000 10^7+1000000000000000000 10^8+100000000000000000 10^9+10000000000000000 10^10+1000000000000000 10^11+100000000000000 10^12+10000000000000 10^13+1000000000000 10^14+100000000000 10^15+10000000000 10^16+1000000000 10^17+100000000 10^18+10000000 10^19+1000000 10^20+100000 10^21+10000 10^22+1000 10^23+100 10^24+10 10^25+10^26)/(10000000000000000000000000+1000000000000000000000000 10+100000000000000000000000 10^2+10000000000000000000000 10^3+1000000000000000000000 10^4+100000000000000000000 10^5+10000000000000000000 10^6+1000000000000000000 10^7+100000000000000000 10^8+10000000000000000 10^9+1000000000000000 10^10+100000000000000 10^11+10000000000000 10^12+1000000000000 10^13+100000000000 10^14+10000000000 10^15+1000000000 10^16+100000000 10^17+10000000 10^18+1000000 10^19+100000 10^20+10000 10^21+1000 10^22+100 10^23+10 10^24+10^25)  =  135/13:
Answer: |  135/13

Apr 16, 2016
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