Question 983102: Please give steps on how to solve these two limit problem, pretty new to these.
lin x ->0 Ln(sin|x|)
lim x -> - ∞ (3+t)/(sqrt(1+9t^2))
Thank yu
Answer by swincher4391(1107)   (Show Source):
You can put this solution on YOUR website! lim x-> 0 ln(sin|x|)
First you want to plug 0 in to see if the point actually exists.
As you can see, we get ln(0), which is undefined.
Usually for trigonometric limits that involve 0, we think of the squeeze theorem.
Let's set up an inequality here. Start with the trig and build it into our function.
-1 < sin(abs(x)) < 1
ln(-1) < ln(sin(abs(x)) < ln(1)
0 < ln(sin(abs(x)) < 0
so lim x-> 0 ln(sin(abs(x)) = 0
lim x-> - ∞ (3+t) / sqrt(1+9t^2)
With this you want to divide everything by the highest degreed term. In this case, that is t.
(3/t + t/t) / (sqrt(1/t^2 + 9t^2/t^2) = (3/t + 1)/sqrt(1/t^2 + 9)
Plugging in infinity gives us
(0+1)/sqrt(0+9) = 1/sqrt(9) = 1/3
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