Evaluate the limit Lim (x+2)/((√x^2-4))x→2
To evaluate the limit of (x + 2) / (√(x^2 – 4)) as x approaches 2, we can use algebraic manipulation and the concept of limits
To evaluate the limit of (x + 2) / (√(x^2 – 4)) as x approaches 2, we can use algebraic manipulation and the concept of limits.
Step 1: Substitute the value of x into the expression.
Let’s substitute x = 2 into (x + 2) / (√(x^2 – 4)):
(2 + 2) / (√(2^2 – 4)) = (4) / (√(4 – 4)) = 4/0
Step 2: Simplify the expression further.
The expression becomes 4/0. This indicates that the denominator is zero, which is undefined in mathematics.
Step 3: Determine the type of limit.
Since the denominator is zero, we have an indeterminate form of 4/0. This means we need to further investigate the limit.
Step 4: Use the conjugate to simplify the expression.
To work around the indeterminate form, we can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator, which is (√(x^2 – 4)) + 2.
[(x + 2) / (√(x^2 – 4))] * [((√(x^2 – 4)) + 2) / ((√(x^2 – 4)) + 2)]
= (x + 2)(√(x^2 – 4) + 2) / (x^2 – 4 + 2√(x^2 – 4) + 4)
= (x + 2)(√(x^2 – 4) + 2) / (x^2 + 2√(x^2 – 4))
Step 5: Substitute the limit value.
Let’s substitute x = 2 into the simplified expression:
(2 + 2)(√(2^2 – 4) + 2) / (2^2 + 2√(2^2 – 4))
= 4(√4 + 2) / (4 + 2√4)
= 4(2 + 2) / (4 + 2 * 2)
= 4(4) / (4 + 4)
= 16 / 8
= 2
Step 6: Interpret the result.
The limit of (x + 2) / (√(x^2 – 4)) as x approaches 2 is equal to 2.
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