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

Question

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

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Answer 1 · 2018-04-02T14:16:49

# 1 / (2sqrt x) #

# 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) #

Answer 2 · 2018-04-02T14:20:22

#1/(2sqrtx)#

#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)#

Answer 3 · 2018-04-02T14:26:23

#L=1/(2sqrtx#

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#