lim x→ infinity of sqrt(x^2 + cx) − sqrt(x^2 + dx)?
Find the limit. (Let c and d represent arbitrary real numbers)
lim x→ infinity of sqrt(x^2 + cx) − sqrt(x^2 + dx)
thanks will vote best answer!
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Find the limit. (Let c and d represent arbitrary real numbers)
lim x→ infinity of sqrt(x^2 + cx) − sqrt(x^2 + dx)
thanks will vote best answer!
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Verified answer
Use conjugates.
lim(x→∞) (√(x^2 + cx) - √(x^2 + dx))
= lim(x→∞) (√(x^2 + cx) - √(x^2 + dx)) * (√(x^2 + cx) + √(x^2 + dx))/(√(x^2 + cx) + √(x^2 + dx))
= lim(x→∞) ((x^2 + cx) - (x^2 + dx))/(√(x^2 + cx) + √(x^2 + dx))
= lim(x→∞) (c - d)x / (√(x^2 + cx) + √(x^2 + dx)).
Now, divide each term by x = √(x^2):
lim(x→∞) (c - d) / (√(1 + c/x) + √(1 + d/x))
= (c - d)/(1 + 1)
= (c - d)/2.
I hope this helps!