First, we'll try to plug in the value: #lim_{x to -oo}x+sqrt(x^2+2x) = -oo + sqrt(oo-oo)# We're already encountering a problem: it is simply not allowed to have #oo-oo#, it's like dividing by zero. We need to try a different approach. Whenever I see this kind of limit, I try to use a trick: #lim_{x to -oo}x+sqrt(x^2+2x)# #= lim_{x to -oo}x+sqrt(x^2+2x)*(x-sqrt(x^2+2x))/(x-sqrt(x^2+2x))# These are the same becaus the factor we're multiplying with is essentially #1#. Why are we doing this? Because there exists a formula which says: #(a-b)(a+b) = a^2-b^2# In this case #a = x# and #b = sqrt(x^2+2x)# Let's apply this formula: #lim_{x to -oo}(x^2-(sqrt(x^2+2x))^2)/(x-sqrt(x^2+2x))# #= lim_{x to -oo}(x^2-x^2-2x)/(x-sqrt(x^2+2x))# #= lim_{x to -oo}(-2x)/(x-sqrt(x^2+2x))# Now we're going to use another trick. We'r going to use this one, because we want to get the #x^2# out of the square root: #lim_{x to -oo}(-2x)/(x-sqrt(x^2(1+2/x))# If you look carefully, you see it's the same thing. Now, you might say that #sqrt(x^2) = x#, but you have to remember that #x# is a negative number. Because we're taking the positive square root, #sqrt(x^2) = -x# in this case. #= lim_{x to -oo}(-2x)/(x+xsqrt(1+2/x))# #= lim_{x to -oo}(-2x)/(x(1+sqrt(1+2/x)))# We can cancel the #x#: #= lim_{x to -oo}(-2)/(1+sqrt(1+2/x))# And now, we can finally plug in the value: #= -2/(1+sqrt(1+2/-oo))# A number divided by infinity, is always #0#: #= -2/(1+sqrt(1+0)) = -2/(1+1) = -2/2 = -1# This is the final answer. Hope it helps.
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