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Evaluate the following limits limit x tends to 0 x + 1 ^ 5 -1/ x

6.   Evaluate the following limits \lim_{x \rightarrow 0 }\frac{( x+1)^5 -1}{x }

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The limit 

\lim_{x \rightarrow 0 }\frac{( x+1)^5 -1}{x }

Lets put 

 x+1=y

since we have changed the function, its limit will also change,

so

x\rightarrow 0,y\rightarrow 0+1=1

So our function have became

\lim_{y \rightarrow 1 }\frac{ y^5 -1}{y-1 }

Now As we know the property 

\lim_{x \rightarrow 1 }\frac{ x^5 -a^n}{x-a }=na^{n-1}

 

\lim_{y \rightarrow 1 }\frac{ y^5 -1}{y-1 }=5(1)^5=5

Hence,

\lim_{x \rightarrow 0 }\frac{( x+1)^5 -1}{x }=5

 

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