# What is the limit? #lim_(theta->0)(tan(3theta^2)+sin^2 5theta)/(theta^2)# and #lim_(x->oo)(cos(pi/x))/(x-2)#

# lim_(theta rarr 0)(tan(3 theta^2)+sin^2 5 theta)/(theta^2) = 28 #

# lim_(x rarr oo)cos(pi/x)/(x-2) = 0#

We can start by using some standard, well defined elementary calculus limits. (all trig function evaluated in radians):

# {: (ul("Limit"), ul("Expression"), ul("Result"), ul("Notes")),

(A, lim_(theta rarr 0) A+B, =lim_(theta rarr 0) A+lim_(theta rarr 0) B,), (B, lim_(theta rarr 0) AB, =lim_(theta rarr 0) A * lim_(theta rarr 0) B,), (C, lim_(theta rarr 0) sin theta, =0,"Direct Evaluation"), (D, lim_(theta rarr 0) cos theta, =1,"Direct Evaluation"), (E, lim_(theta rarr 0) (sin theta)/theta, =1,"Should be learnt"), (F, lim_(theta rarr 0) (1-cos theta)/theta, =0,"Should be learnt"), (G, lim_(theta rarr 0) (tan theta)/theta, =1,) :} #

Limit 1:

We seek:

Limit 2:

We seek:

We can manipulate the limit, in preparation for a substitution:

Both of these limits can be evaluated by direct substitution, thus:

Note 1:

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The limit of (tan(3theta^2)+sin^2 5theta)/(theta^2) as theta approaches 0 is 15.

The limit of (cos(pi/x))/(x-2) as x approaches infinity is 0.

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