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What is the limit as x approaches 0 of #(tan^2x)/(3x)#?

Answer 1

Emilia Aguilar

#lim_(x->0)(tan^2(x))/(3x)=0#

We have:

#lim_(x->0)(tan^2(x))/(3x)#

Since we get #0/0# after substituting #0# in the place of #x#, we can use the L'Hopital's Rule, which states that:

If #(f(c))/(g(c))# gives you an indeterminate form, then #lim_(x->c)=(f'(c))/(g'(c))#

We now have:

#lim_(x->0)(tan^2(x))/(3x)=(d/dx[tan^2(x)])/(d/dx[3x])#

A few things to remember:

#d/dx[tan(x)]=sec^2(x)#

#d/dx[x^n]=nx^(n-1)# If #n# is a constant.

#d/dx[f(g(x))]=f'(g(x))*g'(x)#

#=>(2*tan(x)*d/dx[tan(x)])/(3)#

#=>(2*tan(x)*sec^2(x))/(3)#

We now substitute #0# in the place of #x#.

#=>(2*tan(0)*sec^2(0))/(3)=0#

Therefore, #lim_(x->0)(tan^2(x))/(3x)=0#

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Answer 2

Emily Ammerman

The limit as x approaches 0 of (tan^2x)/(3x) is 1/3.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.