How do you evaluate #sqrt(x 1) /( x1)# as x approaches 1?
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The limit does not exist.
We claim that the limit does not exist.
In view of above, the reqd. limit does not exist.
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To evaluate the limit of sqrt(x  1) / (x  1) as x approaches 1, we can simplify the expression by factoring the denominator.
Since x  1 is a factor of sqrt(x  1), we can cancel out the common factor.
Therefore, the expression simplifies to 1 / sqrt(x  1).
Now, as x approaches 1, the value inside the square root becomes 0.
Taking the square root of 0 gives us 0.
So, the limit of sqrt(x  1) / (x  1) as x approaches 1 is 1 / 0, which is undefined.
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To evaluate ( \frac{\sqrt{x  1}}{x  1} ) as ( x ) approaches 1, we can use algebraic manipulation and limit properties.

Substitute ( x = 1 ) into the expression: [ \frac{\sqrt{x  1}}{x  1} = \frac{\sqrt{1  1}}{1  1} = \frac{0}{0} ]

Recognize that ( \frac{0}{0} ) is an indeterminate form.

To resolve the indeterminate form, rationalize the numerator by multiplying both the numerator and the denominator by the conjugate of the numerator: [ \lim_{x \to 1} \frac{\sqrt{x  1}}{x  1} \cdot \frac{\sqrt{x  1}}{\sqrt{x  1}} = \lim_{x \to 1} \frac{x  1}{(x  1)(\sqrt{x  1})} ]

Simplify: [ \lim_{x \to 1} \frac{x  1}{(x  1)(\sqrt{x  1})} = \lim_{x \to 1} \frac{1}{\sqrt{x  1}} ]

Now, as ( x ) approaches 1, ( \sqrt{x  1} ) approaches 0, and thus ( \frac{1}{\sqrt{x  1}} ) approaches ( \frac{1}{0} ), which is undefined.
Therefore, the limit of ( \frac{\sqrt{x  1}}{x  1} ) as ( x ) approaches 1 does not exist.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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