What is the lim h → 0 (x+h)^1/2-(x)^1/2/h?

lim h rarr 0 (sqrt(x+h)-sqrtx)/h

3 Answers
Apr 2, 2018

1 / (2sqrt x)

Explanation:

lim_(h →0) (sqrt(x+h)-sqrtx)/h

= lim_(h →0) (sqrt(x+h)-sqrtx)/h xx (sqrt(x+h)+sqrtx)/ (sqrt(x+h)+sqrtx)

= lim_(h →0) (sqrt(x+h)^2- sqrtx^2) xx 1/ (hxx(sqrt(x+h)+sqrtx))

= lim_(h →0) ( x+h-x ) xx 1/ (hxx(sqrt(x+0)+sqrtx))

= lim_(h →0) h/ (hxx2sqrt x)

= 1 / (2sqrt x)

Apr 2, 2018

1/(2sqrtx)

Explanation:

lim h->0 (sqrt(x+h)-sqrtx)/h

=lim h->0 (sqrt(x+h)-sqrtx)/h times (sqrt(x+h)+sqrtx)/(sqrt(x+h)+sqrtx)

=lim h->0 (x+h-x)/h times 1/(sqrt(x+h)+sqrtx)

=lim h->0 h/h times 1/(sqrt(x+h)+sqrtx)

=lim h->0 1/(sqrt(x+h)+sqrtx)

=1/(2sqrtx)

Apr 2, 2018

L=1/(2sqrtx

Explanation:

Here,

L=lim_(hto0)(sqrt(x+h)-sqrtx)/h

L=lim_((x+h)tox)((x+h)^(1/2)-x^(1/2))/((x+h)-x)

We know that ,

color(red)(lim_(Xtoa)(X^n-A^n)/(X-A)=nA^(n-1), n inRR)

So,

L=1/2x^(1/2-1)

=>L=1/2x^(-1/2)

=>L=1/(2sqrtx