Find the limit of x squared minus 4 as x approaches positive infinity.

Step-by-step solution

1. 1

   Compare the two terms. As $x$ grows, $x^2$ grows without bound while $-4$ never changes. In any sum, the term with the highest degree eventually dominates, so the limit is governed by $x^2$ alone.

2. 2

   Evaluate the dominant term. $x\to\infty$ gives $x^2\to+\infty$ (squaring a large positive number makes it larger still).

3. 3

   Subtract the constant. Removing a fixed $4$ from an unbounded quantity leaves it unbounded:

   $x^2-4\to+\infty$

   The shift only moves the graph down by $4$ units; it cannot cap its growth.

4. 4

   Verify numerically. At $x=10$: $96$. At $x=100$: $9996$. At $x=1000$: $999996$. Each step of ten in $x$ multiplies the output by roughly $100$.

5. 5

   Interpret the answer honestly. We write

   $\lim_{x\to\infty}(x^2-4)=+\infty$

   but $+\infty$ is not a number: the correct wording is that the limit does not exist as a finite value, and the function diverges to $+\infty$. Note also that $\lim_{x\to-\infty}(x^2-4)=+\infty$ too, since squaring removes the sign.