Question

Lim 2√х+3−4/х−1 (x+3 - под корнем, 2√х+3−4 - числитель, х-1 - знаменатель) x->1

Answer 1

Lim((2√(x+3)-4)/(x-1))=(2*√(1+3)-4)/(1-1)=0/0
x->1
lim((2√(x+3)-4)/(x-1))=
x->1
=lim((2√(x+3)-4)*(2√(x+3)+4))/((x-1)*(2√(x+3)+4))=
x->1
=lim((2√(x+3))²-4²)/((x-1)*(2√(x+3)+4)=
x->1
=lim(4*(x+3)-16)/((x-1)*(2√(x+3)+4))=
x->1
=lim(4x-4)/((x-1)*(2√(x+3)+4))=
x->1
=lim(4*(x-1))/((x-1)*(2(√(x+1)+2))=
x->1
=lim4/2(√(x+3)+2)=lim2/(√(x+3)+2)=2/(√(1+3)+2)=2/4= 1/2=0,5
x->1

Answer 2

To find the limit of the expression as x approaches 1, we can simplify it and evaluate the limit.

First, let's simplify the expression: lim(x->1) (2√(x+3) - 4)/(x-1)

To simplify further, let's rationalize the numerator by multiplying both the numerator and denominator by the conjugate of the numerator, which is (2√(x+3) + 4):

lim(x->1) [(2√(x+3) - 4)/(x-1)] * [(2√(x+3) + 4)/(2√(x+3) + 4)]

This simplifies to:

lim(x->1) (4(x+3) - 16)/(x-1)(2√(x+3) + 4)

Simplifying the numerator:

lim(x->1) (4x + 12 - 16)/(x-1)(2√(x+3) + 4) lim(x->1) (4x - 4)/(x-1)(2√(x+3) + 4) lim(x->1) 4(x - 1)/(x-1)(2√(x+3) + 4)

Now, we can cancel out the common factors:

lim(x->1) 4/(2√(x+3) + 4)

Next, substitute x = 1 into the expression:

lim(x->1) 4/(2√(1+3) + 4) lim(x->1) 4/(2√4 + 4) lim(x->1) 4/(2(2) + 4) lim(x->1) 4/(4 + 4) lim(x->1) 4/8 lim(x->1) 1/2

Therefore, the limit of the expression as x approaches 1 is 1/2.