Evaluate the limit by using a change of variable?

Question

Evelyn Agard · Accepted Answer

Let u = #(x+8)^(1/3)#

Then #u^3=x+8# and #x = u^3-8#

As x approaches the value 0, u approaches the value 2. The given limit becomes

#lim_(x->0) ((x+8)^(1/3)-2)/x = lim_(u->2) (u-2)/(u^3-8)#

#(u-2)/((u-2)(u^2+2u+4))#

As #(u-2)# cancels out and sub 2 in for u provides the final answer of #1/12#

Paisley Johnson · Answer

#1/12#

We observe that in the present form the limit becomes #0/0#. Which is indeterminate.

Therefore, let us substitute #(x+8)^(1/3)=u# #=>x+8=u^3# #=>x = u^3-8#

Also as #x->0#, #u->2#

With this substitution the given question becomes

# lim_(u->2) (u-2)/(u^3-8)#

#=>lim_(u->2)(u-2)/((u-2)(u^2+2u+4))# #=>lim_(u->2)1/((u^2+2u+4))# #=>1/((2^2+2xx2+4))# #=>1/12#

Ezra Adams · Answer

undefined To evaluate a limit using a change of variable, we need to substitute a new variable that simplifies the expression. This technique is particularly useful when dealing with indeterminate forms such as 0/0 or ∞/∞.

To illustrate the process, let's consider an example:

Evaluate the limit as x approaches 0 of (sin(3x))/x.

To simplify this expression, we can introduce a new variable, let's say u, such that u = 3x. This allows us to rewrite the expression as (sin(u))/(u/3).

Now, as x approaches 0, u also approaches 0 since u = 3x. Therefore, we can rewrite the limit as u approaches 0 of (sin(u))/(u/3).

Using the fact that the limit as x approaches 0 of sin(x)/x is equal to 1, we can substitute this result into our expression:

lim(u→0) (sin(u))/(u/3) = (1)/(1/3) = 3.

Therefore, the limit as x approaches 0 of (sin(3x))/x is equal to 3.