Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) asa. x → 0⁺

Accepted Answer

Step 1: Understand the problem. We need to find the limit of the expression $$ \frac{1}{x^{2/3}} + \frac{2}{(x - 1)^{2/3}}  as$$ $$ x 	o 0^+ . $$This means we are approaching 0 from the positive side. Step 2: Analyze the behavior of each term separately as $$ x 	o 0^+ . $$For the term $$ \frac{1}{x^{2/3}} $$, as $$ x $$ approaches 0 from the positive side, $$ x^{2/3} $$ approaches 0, making $$ \frac{1}{x^{2/3}} $$ approach infinity. Step 3: Consider the second term $$ \frac{2}{(x - 1)^{2/3}} . As$$ $$ x 	o 0^+ $$, $$ x - 1 $$ approaches -1. Therefore, $$ (x - 1)^{2/3} $$ approaches $$(-1)^{2/3}$$, which is a real number. Thus, $$ \frac{2}{(x - 1)^{2/3}} $$ approaches a finite value. Step 4: Combine the behavior of both terms. The first term $$ \frac{1}{x^{2/3}} $$ dominates the behavior of the limit because it approaches infinity, while the second term approaches a finite value. Step 5: Conclude the limit. Since the first term approaches infinity and dominates the expression, the limit of the entire expression as $$ x 	o 0^+  is $$infinity.

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.

lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as

a. x → 0⁺

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Key Concepts

Limits

Indeterminate Forms

Behavior Near Zero