Find the limit if it exists.
1. Lim x--> 3 (square root of x+1) -2 over (x-3)
To find the limit of the given expression as x approaches 3, we can directly substitute x=3 into the expression and see if it is defined.
Let's evaluate the expression when x = 3:
lim x->3 [(√(x+1) - 2)/(x-3)]
Substituting x = 3:
= [(√(3+1) - 2)/(3-3)]
= [(√4 - 2)/0]
Since we cannot divide by zero, we need to find another approach to evaluate this limit.
To simplify the expression, we can rationalize the numerator. Multiply the numerator and denominator by the conjugate of the numerator, which is (√(x+1) + 2):
lim x->3 [(√(x+1) - 2)/(x-3)] * [(√(x+1) + 2)/(√(x+1) + 2)]
Simplifying this expression:
= lim x->3 [((√(x+1))^2 - 2^2)/(x-3)(√(x+1) + 2)]
= lim x->3 [(x+1 - 4)/(x-3)(√(x+1) + 2)]
= lim x->3 [(x - 3)/(x-3)(√(x+1) + 2)]
Now, we can cancel out the common term of (x-3):
= lim x->3 [1/(√(x+1) + 2)]
Now, we can substitute x = 3 into this expression:
= 1/(√(3+1) + 2)
= 1/(√4 + 2)
= 1/(2 + 2)
= 1/4
= 0.25
Therefore, the limit of the given expression as x approaches 3 is 0.25.