6.6 *L'Hopital's Rule 355- Use the result of Exercise 2 (b) to prove Cases 13, 14 , and 15 of Theorem
6.6.4. - Calculate each of the following limits. Before using L'Hopital's rule, be
sure tha t the hypotheses are met.
(a) lim^1 - ex (b) lim ln(x/^3 )
x->O X x->3 3 - X
(d) lim ~inx
x->7r s1n4x
2x + 1
(g) lim --
x->O 3x + 1
(j) lim Vx - 1
X->0Vx+1( )^1. sin x
e im --
x->O+ Vx
2x -1
(h) lim --
x->O 3 x - 1
(k) lim ln~x/7r)
X->1r sin x
Prove Case 3 of Theorem 6.6.6.
Prove Cases 4, 5, and 6 of Theorem 6.6.6.
Prove Cases 7, 8, and 9 of Theorem 6.6.6.
Prove Cases 10, 11, and 12 of Theorem 6.6.6.
Prove Cases 13 , 14 , and 15 of Theorem 6.6.6.
(c) lim 1 - cosx
x->O cosx^2
(f) r x cosx
x~ sinx
(i) lim ln (1+1/x)
X-><Xl 1/X(1) lim JXTI -^1
X->0 X
Calculate each of the following limits. Before using L'Hopital's rule, be
sure that the hypotheses are met.
(a) lim ex (b) lim ex
X->-<Xl X
(c) lim lnx (d) lim secx
X->0+ ln X
(e) lim lnx (f) lim 3 + 4secx
x->O+ ex X->7r / 2- 2 + tan X
(g) lim lnsin2x (h) lim 1/(x - 2)
X->7r/ 2 - lnCOSX X->2+ ln(x-2)
Calculate each of the following limits. In each case, describe the inde-
terminate form, and transform it into a form to which L 'Hopital's rule
applies. (Before using L'Hopital's rule, be sure the hypotheses are met.)
(a) lim (secx - t anx) (b) lim (~ + lnx)
X->7r /2+ x->O+
(c) lim x ln(sinx) (d) lim xsinx
x->O+ x->O+
(e) lim (1-x)^11 x (f) lim (1 - ~r
x--+O+ x--+oo x
(g) lim (ex+ x)^1 /x (h) lim (.!_ - -
1
-)
X-><Xl X->0 X eX - 1