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

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 <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> approaches <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Evaluate the limit of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> which is constant as <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> approaches <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Simplify the expression.

Evaluate the limit of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by plugging in <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Evaluate the limit of the denominator.

Evaluate the limit.

Split the limit using the Sum of Limits Rule on the limit as <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> approaches <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Move the limit under the radical sign.

Split the limit using the Sum of Limits Rule on the limit as <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> approaches <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Evaluate the limit of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> which is constant as <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> approaches <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Evaluate the limit of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> which is constant as <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> approaches <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Evaluate the limit of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by plugging in <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Simplify the answer.

Simplify each term.

Add <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Pull terms out from under the radical, assuming positive real numbers.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

The expression contains a division by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> . The expression is undefined.

Undefined

The expression contains a division by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> . The expression is undefined.

Undefined

The expression contains a division by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> . The expression is undefined.

Undefined

Since <math><mstyle displaystyle="true"><mfrac><mrow><mn>0</mn></mrow><mrow><mn>0</mn></mrow></mfrac></mstyle></math> 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 <math><mstyle displaystyle="true"><mn>2</mn><mo>-</mo><mi>x</mi></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mn>2</mn><mo>]</mo></mrow><mo>+</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mo>-</mo><mi>x</mi><mo>]</mo></mrow></mstyle></math> .

Since <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> is constant with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> , the derivative of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mo>-</mo><mi>x</mi><mo>]</mo></mrow></mstyle></math> .

Since <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> is constant with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> , the derivative of <math><mstyle displaystyle="true"><mo>-</mo><mi>x</mi></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow></mstyle></math> .

Differentiate using the Power Rule which states that <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mi>n</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi><mo>=</mo><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

By the Sum Rule, the derivative of <math><mstyle displaystyle="true"><msqrt><mi>x</mi><mo>+</mo><mn>2</mn></msqrt><mo>-</mo><mn>2</mn></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><msqrt><mi>x</mi><mo>+</mo><mn>2</mn></msqrt><mo>]</mo></mrow><mo>+</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mo>-</mo><mn>2</mn><mo>]</mo></mrow></mstyle></math> .

Evaluate <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><msqrt><mi>x</mi><mo>+</mo><mn>2</mn></msqrt><mo>]</mo></mrow></mstyle></math> .

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mi>x</mi><mo>+</mo><mn>2</mn></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Differentiate using the chain rule, which states that <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mi>f</mi><mrow><mo>(</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>]</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mi>f</mi><mo>′</mo><mrow><mo>(</mo><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>)</mo></mrow><mi>g</mi><mo>′</mo><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mstyle></math> where <math><mstyle displaystyle="true"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> and <math><mstyle displaystyle="true"><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mi>x</mi><mo>+</mo><mn>2</mn></mstyle></math> .

To apply the Chain Rule, set <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> as <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>2</mn></mstyle></math> .

Differentiate using the Power Rule which states that <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>u</mi></mrow></mfrac><mrow><mo>[</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mi>n</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Replace all occurrences of <math><mstyle displaystyle="true"><mi>u</mi></mstyle></math> with <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>2</mn></mstyle></math> .

By the Sum Rule, the derivative of <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>2</mn></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mi>x</mi><mo>]</mo></mrow><mo>+</mo><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><mn>2</mn><mo>]</mo></mrow></mstyle></math> .

Differentiate using the Power Rule which states that <math><mstyle displaystyle="true"><mfrac><mrow><mi>d</mi></mrow><mrow><mi>d</mi><mi>x</mi></mrow></mfrac><mrow><mo>[</mo><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>]</mo></mrow></mstyle></math> is <math><mstyle displaystyle="true"><mi>n</mi><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi><mo>-</mo><mn>1</mn></mrow></msup></mstyle></math> where <math><mstyle displaystyle="true"><mi>n</mi><mo>=</mo><mn>1</mn></mstyle></math> .

Since <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> is constant with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> , the derivative of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

To write <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Move the negative in front of the fraction.

Add <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow><mrow><mn>2</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Move <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mo>-</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> to the denominator using the negative exponent rule <math><mstyle displaystyle="true"><msup><mrow><mi>b</mi></mrow><mrow><mo>-</mo><mi>n</mi></mrow></msup><mo>=</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msup><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac></mstyle></math> .

Since <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> is constant with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> , the derivative of <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Multiply the numerator by the reciprocal of the denominator.

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><msqrt><mi>x</mi><mo>+</mo><mn>2</mn></msqrt></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> .

Move the term <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> outside of the limit because it is constant with respect to <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Move the limit under the radical sign.

Evaluate the limit of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by plugging in <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Pull terms out from under the radical, assuming positive real numbers.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

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