Calculate the limit of the fraction (f(x)-f(1))/(x-1) if f(x)=x^300+x+1, x-->1

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We notice that if we'll calculate the limit of the given ratio, we'll calculate the first derivative of the given function, for x = 1.

lim (f(x)-f(1))/(x-1) = f'(1)

For this reason, we'll calculate first the derivative of the function:

f'(x) = 300x + 1

Now, to evaluate f'(1), we'll substitute x by 1 in the expression of derivative:

f'(1) = 300*1 + 1

f'(1) = 301

So, it is no need to struggle calculating the limit of the ratio, when we can do an easier way:

**lim [(f(x)-f(1))/(x-1)] = 301 for x-> 1**

We have f(x) = x^300 + x + 1 and we have to find

lim x-->1 [ (f(x)-f(1))/(x-1)]

=> lim x-->1 [ x^300 + x + 1 - 3)/ (x-1)]

for x = 1 we have the form 0/0, so we can use l'hopital's rule

=> lim x-->1 [ 300*x^299 + 1 )]

for x = 1, we have 300*1 + 1 = 301

**The required limit of the function is 301**

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