How do you find #lim t(sqrt(t+1)-sqrtt)# as #t->oo#?

Answer 1

#lim_(t->oo) t(sqrt(t+1)-sqrt(t)) = oo#

#lim_(t->oo) t(sqrt(t+1)-sqrt(t))#
#=lim_(t->oo) (t(sqrt(t+1)-sqrt(t))(sqrt(t+1)+sqrt(t)))/(sqrt(t+1)+sqrt(t))#
#=lim_(t->oo) (t((t+1)-t))/(sqrt(t+1)+sqrt(t))#
#=lim_(t->oo) t/(sqrt(t+1)+sqrt(t))#
#=lim_(t->oo) sqrt(t) * sqrt(t)/(sqrt(t+1)+sqrt(t))#
#=lim_(t->oo) sqrt(t) * 1/(sqrt(1+1/t)+1)#
#=lim_(t->oo) 1/2sqrt(t)#
#=oo#
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Answer 2

#+oo#

I'll replace #t# with #x#
#lim_(xrarr+oo)x*(sqrt(x+1)-sqrtx)# #=#
#lim_(xrarr+oo)x*((sqrt(x+1)-sqrtx)(sqrt(x+1)+sqrtx))/(sqrt(x+1)+sqrtx)# #=#
#lim_(xrarr+oo)x*(sqrt(x+1)^2-sqrtx^2)/(sqrt(x+1)+sqrtx)# #=#
#lim_(xrarr+oo)x*(cancel(x)+1-cancel(x))/(sqrt(x+1)+sqrtx)# #=#
#lim_(xrarr+oo)x/(sqrt(x+1)+sqrtx)# #=#
#lim_(xrarr+oo)x/(sqrt(x^2(1/x+1/x^2))+sqrt(x^2*(1/x)))# #=#
#lim_(xrarr+oo)x/(|x|sqrt(1/x+1/x^2)+|x|sqrt(1/x))#
#x->+oo# , #x>0#
#=# #lim_(xrarr+oo)x/(xsqrt(1/x+1/x^2)+xsqrt(1/x))# #=#
#lim_(xrarr+oo)cancel(x)/(cancel(x)(sqrt(1/x+1/x^2)+sqrt(1/x))# #=#
#lim_(xrarr+oo)1/(sqrt(1/x+1/x^2)+sqrt(1/x))# #=# #+oo#
because #lim_(xrarr+oo)x=+oo# so #lim_(xrarr+oo)1/x=0#
and #lim_(xrarr+oo)sqrt(1/x)=0#
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Answer 3

To find the limit of the expression lim t(sqrt(t+1)-sqrt(t)) as t approaches infinity, we can use algebraic manipulation and the properties of limits.

First, let's simplify the expression by rationalizing the denominator. Multiply the expression by the conjugate of the denominator, which is sqrt(t+1) + sqrt(t):

lim t(sqrt(t+1)-sqrt(t)) * (sqrt(t+1)+sqrt(t))/(sqrt(t+1)+sqrt(t))

Expanding the numerator and denominator, we get:

lim t((sqrt(t+1))^2 - (sqrt(t))^2)/(sqrt(t+1)+sqrt(t))

Simplifying further:

lim t(t+1 - t)/(sqrt(t+1)+sqrt(t))

lim t(1)/(sqrt(t+1)+sqrt(t))

As t approaches infinity, the denominator sqrt(t+1) + sqrt(t) also approaches infinity. Therefore, the limit of the expression is:

lim t(sqrt(t+1)-sqrt(t)) = 0

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Answer from HIX Tutor

When evaluating a one-sided 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 one-sided 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 one-sided 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 one-sided 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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