lim (x^(2)-4)/(x-2)
x->2
lim
x->1 (x+1)/(x-1)
( x ^ 2 - 4 ) / ( x - 2 ) =
( x + 2 ) * ( x - 2 ) / ( x - 2 ) =
x + 2
lim ( x ^ 2 - 4 ) / ( x - 2 ) as x->2 =
lim ( x + 2 ) as x->2 = 2 + 2 = 4
( x + 1 )/ ( x - 1 ) = 2 / ( x - 1 ) + 1
lim ( x + 1 )/ ( x - 1 ) as x->1 =
lim [ 2 / ( x - 1 ) + 1 ] as x->1 = + OR - infinitty
becouse lim 2 / ( x - 1 ) as x->1 = + OR - infinity
To find the limit of a function as x approaches a specific value, we first substitute that value into the function and simplify. Let's calculate the limits you mentioned step by step.
1. Lim(x->2) (x^2 - 4)/(x - 2):
To find this limit, we can directly substitute x = 2 into the equation and evaluate:
(x^2 - 4)/(x - 2) = (2^2 - 4)/(2 - 2) = (4 - 4)/(0) = 0/0
Here, we obtain an indeterminate form of 0/0, which means we cannot directly evaluate the limit by substitution. To proceed, we need to simplify the expression and apply algebraic manipulation.
By factoring, we can rewrite the expression as:
(x^2 - 4)/(x - 2) = ((x - 2)(x + 2))/(x - 2)
Now, we can cancel out the common factor of (x - 2):
lim(x->2) ((x - 2)(x + 2))/(x - 2) = lim(x->2) (x + 2)
Finally, we substitute x = 2 into the simplified expression:
lim(x->2) (x + 2) = 2 + 2 = 4
Therefore, the limit as x approaches 2 of (x^2 - 4)/(x - 2) is equal to 4.
2. Lim(x->1) (x + 1)/(x - 1):
Again, we substitute x = 1 directly into the equation:
(x + 1)/(x - 1) = (1 + 1)/(1 - 1) = 2/0
Here, we obtain an indeterminate form of 2/0. To simplify further and evaluate the limit, we need to apply algebraic manipulation.
By factoring the numerator, we get:
(x + 1)/(x - 1) = (1(x + 1))/(x - 1)
Now, we can cancel out the common factor of (x - 1):
lim(x->1) (1(x + 1))/(x - 1) = lim(x->1) (x + 1)
Finally, substituting x = 1 into the simplified expression:
lim(x->1) (x + 1) = 1 + 1 = 2
Therefore, the limit as x approaches 1 of (x + 1)/(x - 1) is equal to 2.