Find the limit. lim 1/x^2-4
a)x->2^+
b)x->2^-
c)x->-2^+
d)x->-2^-
I'm completely lost. I don't understand how some answers are infinity and the others are negative infinity.
well since the fraction approaches 1/0, clearly it will approach either +∞ or -∞
So, just check the denominator to see whether it is positive or negative.
a) x->2+ means x > 2. So, x^2 > 4 and the limit is +∞
similarly for the others.
To find the limit of a given expression, we need to analyze the behavior of the expression as x approaches a particular value. Let's evaluate the given limit step by step for each option:
a) x -> 2^+ (approaching 2 from the right)
To find this limit, substitute x = 2 into the expression: lim (1/(x^2 - 4)) as x approaches 2 from the right.
lim (1/(2^2 - 4)) = lim (1/(4 - 4)) = lim (1/0) = infinity (since dividing by zero yields infinity)
b) x -> 2^- (approaching 2 from the left)
To find this limit, substitute x = 2 into the expression: lim (1/(x^2 - 4)) as x approaches 2 from the left.
lim (1/(2^2 - 4)) = lim (1/(4 - 4)) = lim (1/0) = infinity (since dividing by zero yields infinity)
c) x -> -2^+ (approaching -2 from the right)
To find this limit, substitute x = -2 into the expression: lim (1/(x^2 - 4)) as x approaches -2 from the right.
lim (1/((-2)^2 - 4)) = lim (1/(4 - 4)) = lim (1/0) = infinity (since dividing by zero yields infinity)
d) x -> -2^- (approaching -2 from the left)
To find this limit, substitute x = -2 into the expression: lim (1/(x^2 - 4)) as x approaches -2 from the left.
lim (1/((-2)^2 - 4)) = lim (1/(4 - 4)) = lim (1/0) = infinity (since dividing by zero yields infinity)
In this case, all four options yield an answer of infinity because the expression has a vertical asymptote at x = -2 and x = 2, where the denominator becomes zero. As x approaches these values from either side, the expression tends towards infinity.