Consider the function (4)/(1-2^(1/x))
1. Estimate the right-hand limit, lim x->0+ f(x). Back up your answer by making an analytic argument about f(x) for values of x that are close to 0 but larger than 0.
2. Estimate the left-hand limit, lim x->0- f(x). Back up your estimate by making an analytic argument about f(x) for values of x that are close to 0 but smaller than 0.
Looking at the graph at
http://www.wolframalpha.com/input/?i=%284%29%2F%281-2^%281%2Fx%29%29+
I'm sure you can justify your analysis.
Does it have an undefined limit.
correct. The limit from the right is not the same as the limit from the left.
But both limits are finite.
To estimate the right-hand limit, lim x->0+ f(x), we need to analyze the behavior of f(x) for values of x that are close to 0 but larger than 0.
1. For x > 0, let's examine the expression (1-2^(1/x)) first. As x approaches 0 from the right (x->0+), the term 2^(1/x) becomes very large because the exponent approaches infinity. Since 2 raised to a large positive exponent results in a large positive number, we have 2^(1/x) -> ∞ as x -> 0+.
2. Considering the denominator of the function, 1-2^(1/x), as x approaches 0 from the right, the expression becomes 1 - ∞. This expression tends towards negative infinity (1 - ∞ -> -∞).
3. Now, let's focus on the entire function f(x) = 4 / (1 - 2^(1/x)). As the denominator approaches negative infinity (1 - ∞), the fraction 4 / (1 - ∞) becomes very close to zero since the denominator is becoming extremely large and negative.
Based on the analysis above, we can conclude that as x approaches 0 from the right (x -> 0+), the function f(x) approaches 0.
Now, let's estimate the left-hand limit, lim x->0- f(x), by analyzing the behavior of f(x) for values of x that are close to 0 but smaller than 0.
1. For x < 0, we need to consider the expression (1-2^(1/x)). As x approaches 0 from the left (x->0-), the term 2^(1/x) becomes very large again because the exponent approaches infinity. Since 2 raised to a large negative exponent results in a positive number, we have 2^(1/x) -> ∞ as x -> 0-.
2. Considering the denominator of the function, 1-2^(1/x), as x approaches 0 from the left, the expression becomes 1 - ∞. This expression tends towards positive infinity (1 - ∞ -> +∞).
3. Analyzing the entire function f(x) = 4 / (1 - 2^(1/x)), as the denominator approaches positive infinity (1 - ∞), the fraction 4 / (1 - ∞) becomes very close to zero since the denominator is becoming extremely large and positive.
Based on the analysis above, we can conclude that as x approaches 0 from the left (x -> 0-), the function f(x) also approaches 0.
Hence, the estimations for the right-hand limit, lim x->0+ f(x), and the left-hand limit, lim x->0- f(x), are both 0.