How do you evaluate #sqrt(x -1) /( x-1)# as x approaches 1?

Question

Christopher Agresta · Accepted Answer

#=lim_(x->1) (sqrt(x - 1))/(x - 1) xx (sqrt(x - 1))/(sqrt(x - 1))# #=lim_(x->1) (x - 1)/(x - 1 xx sqrt(x - 1))# #=lim_(x->1)1/(sqrt(x - 1))# If you look at the limit from both sides, the quantities are unequal, and the function is discontinuous at the point #x = 1#. Hopefully this helps!

Isabella Ackerman · Answer

The limit does not exist.

We claim that the limit does not exist. To see this, observe that #sqrt(x-1)/(x-1)=1/sqrt(x-1)#. Now, as #xrarr1+, x>1, (x-1)rarr0+, sqrt(x-1)rarr0+# so that,#1/sqrt(x-1)rarroo# But, as #xrarr1-,x<1,(x-1)<0, sqrt(x-1)# is not defined. In view of above, the reqd. limit does not exist.

Charles Anctil · Answer

undefined 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.

Elijah Abram · Answer

undefined To evaluate $ \frac{\sqrt{x - 1}}{x - 1} $ as $ x $ approaches 1, we can use algebraic manipulation and limit properties.

1. Substitute $ x = 1 $ into the expression:
$$ \frac{\sqrt{x - 1}}{x - 1} = \frac{\sqrt{1 - 1}}{1 - 1} = \frac{0}{0} $$

2. Recognize that $ \frac{0}{0} $ is an indeterminate form.

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

4. Simplify:
$$ \lim_{x \to 1} \frac{x - 1}{(x - 1)(\sqrt{x - 1})} = \lim_{x \to 1} \frac{1}{\sqrt{x - 1}} $$

5. 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.