Evaluate the limit of the function (x^2-1)/(x-1) if x goes to 1?
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We'll substitute x by 1 in the expression of the function:
lim (x^2 - 1)/(x-1) = (1-1)/(1-1) = 0/0
Since we've get an indetermination, we can apply L'Hospital's rule:
lim f(x)/g(x) = lim f'(x)/g'(x)
Let f(x) = x^2 - 1 => f'(x) = 2x
Let g(x) = x - 1 => g'(x) = 1
lim (x^2 - 1)/(x-1) = lim 2x/1
We'll replace x by 1:
lim 2x/1 = 2/1 = 2
The requested limit of the given function, if x approaches to 1, is lim (x^2 - 1)/(x-1) = 2.
Since `(x^2-1) = (x-1)(x+1)`
`lim_(x->1) (x^2-1)/(x-1) = lim_(x->1) ((x+1)(x-1))/(x-1) = lim_(x->1) (x+1) = 2`
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