How do you find the limit
#lim_(h-0)(sqrt(1+h)-1)/h# ?

Question

How do you find the limit
#lim_(h->0)(sqrt(1+h)-1)/h# ?

William Abdalla · Accepted Answer

#\frac{1}{2}#

The limit presents an undefined form #0/0#. In this case, you may use de l'hospital theorem, that states

#lim \frac {f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)}#

The derivative of the numerator is

#\frac{1}{2sqrt(1+h)}#

While the derivative of the denominator is simply #1#.

So,

#\lim_{x\to 0} \frac{f'(x)}{g'(x)} = \lim_{x\to 0} \frac{\frac{1}{2sqrt(1+h)}}{1} =\lim_{x\to 0} \frac{1}{2sqrt(1+h)}#

And thus simply

#\frac{1}{2sqrt(1)}=\frac{1}{2}#

Luke Alvarez · Answer

# = 1/2 #

If you are unaware of l'hopitals rule...

Use:

#(1+x)^n = 1 + nx + (n(n-1))/(2!) x^2 + ... #

#=> (1 + h)^(1/2) = 1 + 1/2h - 1/8 h^2 + ... #

#=> lim_( h to 0) ((1 + 1/2 h - 1/8h^2 + ...) - 1 )/ h #

#=> lim_( h to 0) ( 1/2 h - 1/8h^2 + ... )/ h #

#=> lim_( h to 0) ( 1/2 - 1/8 h + ... ) #

# = 1/2 #

Charles Anctil · Answer

undefined To find the limit of the given expression, we can use algebraic manipulation and the concept of limits.

First, let's simplify the expression by multiplying the numerator and denominator by the conjugate of the numerator, which is sqrt(1+h) + 1.

This gives us:
[(sqrt(1+h) - 1) * (sqrt(1+h) + 1)] / [h * (sqrt(1+h) + 1)]

Expanding the numerator using the difference of squares, we get:
[(1+h) - 1] / [h * (sqrt(1+h) + 1)]

Simplifying further, we have:
h / [h * (sqrt(1+h) + 1)]

Now, we can cancel out the h terms:
1 / (sqrt(1+h) + 1)

Finally, we can take the limit as h approaches 0:
lim_(h->0) (1 / (sqrt(1+h) + 1)) = 1 / (sqrt(1+0) + 1) = 1 / (1 + 1) = 1/2

Therefore, the limit of the given expression as h approaches 0 is 1/2.

Emilia Aguilar · Answer

undefined To find the limit of $\lim_{h \to 0} \frac{\sqrt{1+h} - 1}{h}$, you can use the concept of derivatives or L'Hôpital's Rule. Alternatively, you can rationalize the expression to simplify it and then find the limit.

How do you find the limit #lim_(h-&gt;0)(sqrt(1+h)-1)/h# ?

How do you find the limit #lim_(h->0)(sqrt(1+h)-1)/h# ?