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.
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Evaluate the limit of the numerator and the limit of the denominator.
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Take the limit of the numerator and the limit of the denominator.
Evaluate the limit of the numerator.
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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.
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Evaluate the limit of by plugging in for .
Subtract from .
Evaluate the limit of the denominator.
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Evaluate the limit.
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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 the answer.
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Simplify each term.
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Add and .
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.
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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 .
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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 .
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Use to rewrite as .
Differentiate using the chain rule, which states that is where and .
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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.
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Multiply by .
Subtract from .
Move the negative in front of the fraction.
Add and .
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 .
Add and .
Multiply the numerator by the reciprocal of the denominator.
Rewrite as .
Multiply by .
Move the term outside of the limit because it is constant with respect to .
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 by plugging in for .
Add and .
Rewrite as .
Pull terms out from under the radical, assuming positive real numbers.
Multiply by .
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Name

Name one billion six hundred thirty-two million three hundred forty-four thousand four hundred sixteen

Interesting facts

  • 1632344416 has 128 divisors, whose sum is 13048021080
  • The reverse of 1632344416 is 6144432361
  • Previous prime number is 19

Basic properties

  • Is Prime? no
  • Number parity even
  • Number length 10
  • Sum of Digits 34
  • Digital Root 7

Name

Name four hundred ninety-eight million six hundred seventy-five thousand five hundred twenty-three

Interesting facts

  • 498675523 has 4 divisors, whose sum is 501259528
  • The reverse of 498675523 is 325576894
  • Previous prime number is 193

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 9
  • Sum of Digits 49
  • Digital Root 4

Name

Name one billion five hundred seventy-one million eight hundred twenty-five thousand three hundred sixty-three

Interesting facts

  • 1571825363 has 4 divisors, whose sum is 1572209808
  • The reverse of 1571825363 is 3635281751
  • Previous prime number is 4133

Basic properties

  • Is Prime? no
  • Number parity odd
  • Number length 10
  • Sum of Digits 41
  • Digital Root 5