# Write the Fraction in Simplest Form limit as x approaches 2 of (2-x)/( square root of x+2-2)

Write the Fraction in Simplest Form limit as x approaches 2 of (2-x)/( square root of x+2-2)
Apply L'Hospital's rule.
Evaluate the limit of the numerator and the limit of the denominator.
Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Evaluate the limit of which is constant as approaches .
Simplify the expression.
Evaluate the limit of by plugging in for .
Subtract from .
Evaluate the limit of the denominator.
Evaluate the limit.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Move the limit under the radical sign.
Split the limit using the Sum of Limits Rule on the limit as approaches .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of which is constant as approaches .
Evaluate the limit of by plugging in for .
Simplify each term.
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
Subtract from .
The expression contains a division by . The expression is undefined.
Undefined
The expression contains a division by . The expression is undefined.
Undefined
The expression contains a division by . The expression is undefined.
Undefined
Since is of indeterminate form, apply L'Hospital's Rule. L'Hospital's Rule states that the limit of a quotient of functions is equal to the limit of the quotient of their derivatives.
Find the derivative of the numerator and denominator.
Differentiate the numerator and denominator.
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Subtract from .
By the Sum Rule, the derivative of with respect to is .
Evaluate .
Use to rewrite as .
Differentiate using the chain rule, which states that is where and .
To apply the Chain Rule, set as .
Differentiate using the Power Rule which states that is where .
Replace all occurrences of with .
By the Sum Rule, the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
To write as a fraction with a common denominator, multiply by .
Combine and .
Combine the numerators over the common denominator.
Simplify the numerator.
Multiply by .
Subtract from .
Move the negative in front of the fraction.
Combine and .
Multiply by .
Move to the denominator using the negative exponent rule .
Since is constant with respect to , the derivative of with respect to is .