Question

limit x tends to 4 √(x+5) -3 ÷ x-4​

Answer 1

so now let see one example
what is the the limit of
$\frac{ \sqrt{x + 5 } - 3}{x - 4}$

as x approaches 4
sol .
let's multiplying both side numerator and denominator of the expression by
$\sqrt{x + 5} + 3$
to get rid to undefined
$\frac{0}{0}$
value
assuming is not equal to 4
then
$\frac{ \sqrt{x + 5} - 3}{x - 4} =$
$= \frac{ (\sqrt{x + 5} - 3).( \sqrt{x + 5} + 3}{(x - 4).( \sqrt{x + 5} + 3)}$
$= \frac{x + 5 - 9}{{(x - 4).( \sqrt{x + 5} + 3)} \: } = \frac{x - 4}{{(x - 4).( \sqrt{x + 5} + 3)} \: }$

$\frac{1}{( \sqrt{x + 5} + 3) }$
as x→4 this expression tends to
$\frac{1}{ \sqrt{4 + 5} + 3} = \frac{1}{6}$
therefore
lim x → 4
$\frac{ \sqrt{x + 5} - 3 }{x - 4} = \frac{1}{6}$
by this presses you can get answer

Answer 2

HEY THERE!!

lim {√(x+5) -3}/x-4

x->4

rastionalising it we get,

lim(√(x+5) -3)/(x-4 )× (√(x+5) +3)/(√(x+5)+3)

x->4

lim( (x+5) - 3²)/(x-4)((√x+5) +3)

x->4

lim (x+5 -9)/(x-4)((√x+5)+3)

x->4

lim (x-4)/(x-4)((√x+5)+3)

x->4

lim 1/(√(x+5)+3)

x->4

now put x = 4 and the limiting value is

=> 1/(√4+5)+3)

=> 1/6

the limiting value is 1/6

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#Thanks