How do you find the limit #(1+5/sqrtx)/(2+1/sqrtx)# as #x-0^+#?

Question

How do you find the limit #(1+5/sqrtx)/(2+1/sqrtx)# as #x->0^+#?

Evelyn Agard · Accepted Answer

# lim_(x->0^+)((1+5/sqrt(x))/(2+1/sqrt(x))) = 5#

If we look at the graph of #y=(1+5/sqrt(x))/(2+1/sqrt(x))# we can see that it is clear that the limit exists, and is approximately #5#

graph{(1+5/sqrt(x))/(2+1/sqrt(x)) [-8, 8, -2, 10]}

Now, As #x->0# then #1/x->oo# and #1/sqrtx->oo# but if we can invert these expressions they both #->0#

So, we look for a way to invert the #1/A# expression

# lim_(x->0^+)((1+5/sqrt(x))/(2+1/sqrt(x))) = lim_(x->0^+)sqrtx/sqrtx * (1+5/sqrt(x))/(2+1/sqrt(x)) #

# :. lim_(x->0^+)((1+5/sqrt(x))/(2+1/sqrt(x))) = ((lim_(x->0^+)sqrtx)+5) / ((lim_(x->0^+)sqrtx)+1)#

# :. lim_(x->0^+)((1+5/sqrt(x))/(2+1/sqrt(x))) = (0+5)/(0+1)#

# :. lim_(x->0^+)((1+5/sqrt(x))/(2+1/sqrt(x))) = 5#

Which is completely consistent with the above graph.

Elijah Abram · Answer

undefined To find the limit of (1+5/sqrt(x))/(2+1/sqrt(x)) as x approaches 0 from the positive side, we can simplify the expression by multiplying the numerator and denominator by sqrt(x). This gives us (sqrt(x) + 5)/(2sqrt(x) + 1).

Now, as x approaches 0 from the positive side, sqrt(x) also approaches 0. Therefore, we can substitute 0 for sqrt(x) in the simplified expression.

Doing so, we get (0 + 5)/(2*0 + 1), which simplifies to 5/1 or simply 5.

Therefore, the limit of (1+5/sqrt(x))/(2+1/sqrt(x)) as x approaches 0 from the positive side is 5.

How do you find the limit #(1+5/sqrtx)/(2+1/sqrtx)# as #x-&gt;0^+#?

How do you find the limit #(1+5/sqrtx)/(2+1/sqrtx)# as #x->0^+#?