Find the limit of lim (x --> 0) sin(3x) / (5x^3 - 4x)

Question

Find the limit of 

 

lim (x --> 0)     sin(3x) / (5x³ - 4x)

 

 

ANSWER: - 3/4

#43, p 197

Answer 1

We can also use L'Hospital Rule.  If you haven't learned it yet, today's your lucky day.  You get a head start in your class.  

 

We use this rule when you plug in an x value to evaluate the limit, and the limit is indeterminate for example 0/0 , ∞/∞, or the denominator becomes zero.  When this happens, we take the derivative of the numerator over the derivative of the denominator.  So using L'Hospital Rule, we end up with

 

 

lim       3cos(3x) / (15x² - 4)

x-->0

 

 

Now we can plug in x=0 to evaluate the limit.

 

lim           (3 * 1) / (0 - 4)

x-->0

 

 

lim        -3 / 4

x-->0

 

 

As x approaches 0, the limit is -3/4.

 

Answer 2

lim (x->0) sin(3x) / (5x³ - 4x) => factor x in denominator:

 

= lim (x->0) sin(3x) /[x(5x² - 4)] => rewrite as:

 

= lim (x->0) sin(3x)/(3x) * [3/(5x² - 4)] => use: lim(x->0) sin(x)/x = 1:

 

= lim(x->0) 1 * [3/(5x² - 4)] => apply the limit:

 

= [3/(0 - 4)]

 

= -3/4