1.) Lim [√(x + 1) - (2)] / (x - 3)
x -> 3
2.) Lim [ (1/ x + 4) - (1 / 4)] / (x)
x -> 0
Take the ratio of the derivatives of numerator and denominator, and evaluate it at the x value in question.
For both problems, the derivative of the denominator is just 1, so you just have to evaluate the derivative of the numerator.
In the first problem,
Lim [�ã(x + 1)-(2)]/(x-3)
x -> 3
= (1/2�ã(x + 1)
= 1/4 at x =3
For the second problem
Lim [(1/(x+4)-(1/4)]/ x
x -> 0
= -1(x+4)^2 at x=0
= -1/16
To find the limit of a function as x approaches a certain value, you can either directly substitute the value (if it doesn't result in an indeterminate form) or use limit laws to simplify the expression.
1.) Lim [√(x + 1) - 2] / (x - 3)
x -> 3
In this case, directly substituting x = 3 would result in a 0/0 indeterminate form, so we need to simplify the expression using limit laws.
Step 1: Rationalize the numerator
Multiply the numerator and denominator by the conjugate of the numerator, which is √(x + 1) + 2, to eliminate the square root.
Lim [√(x + 1) - 2] * [√(x + 1) + 2] / (x - 3) * [√(x + 1) + 2]
x -> 3
Step 2: Simplify the expression
You can now simplify the denominator by using the difference of squares, (a^2 - b^2) = (a - b)(a + b).
Lim [(x + 1) - 2^2] / [(x - 3) * (√(x + 1) + 2)]
x -> 3
Lim (x - 3) / [(x - 3) * (√(x + 1) + 2)]
x -> 3
Step 3: Cancel out common factors
Cancel out the common factor (x - 3) in the numerator and denominator.
Lim 1 / (√(x + 1) + 2)
x -> 3
Step 4: Substitute the limit value for x
Now that the expression is in a simplified form, you can substitute x = 3 directly into the expression.
Lim 1 / (√(3 + 1) + 2)
x -> 3
Lim 1 / (√4 + 2)
x -> 3
Lim 1 / (2 + 2)
x -> 3
Lim 1 / 4
x -> 3
The limit of the given function as x approaches 3 is 1/4.
2.) Lim [(1 / (x + 4)) - (1 / 4)] / x
x -> 0
In this case, directly substituting x = 0 in the expression would result in an indeterminate form of (1/0) - (1/4), which is undefined. Therefore, we need to simplify the expression.
Step 1: Combine the fractions
Combine the two fractions by finding the least common denominator (LCD), which in this case is 4(x + 4), then perform the subtraction.
Lim [(4 - (x + 4)) / (4(x + 4))] / x
x -> 0
Lim [(4 - x - 4) / (4(x + 4))] / x
x -> 0
Simplify the numerator:
Lim (-x / (4(x + 4))) / x
x -> 0
Step 2: Simplify the expression further
Now you can simplify the expression by canceling out common factors.
Lim (-1 / (4(x + 4)))
x -> 0
Step 3: Substitute the limit value for x
Now that the expression is in a simplified form, you can substitute x = 0 directly into the expression.
Lim (-1 / (4(0 + 4)))
x -> 0
Lim (-1 / 16)
x -> 0
The limit of the given function as x approaches 0 is -1/16.