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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justaguide's profile pic

Posted on

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

giorgiana1976's profile pic

Posted on

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

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