Question

Evaluate the following limit: \[ \lim_{t \to \infty} \sqrt{t^2 + t - t} \]

Hint:

When evaluating limits of expressions involving square roots, consider factoring out the highest power term from both the numerator and denominator if applicable. In this case, factor \( t^2 \) from the square root expression to simplify the problem, and use approximations for large \( t \) to find the limit.

Answer 1

We begin by simplifying the expression inside the square root: \[ \sqrt{t^2 + t - t} = \sqrt{t^2} \] For large \( t \), we can approximate: \[ \sqrt{t^2 + t - t} = \sqrt{t^2(1 + \frac{1}{t})} \] Using the binomial expansion for \( \sqrt{1 + \frac{1}{t}} \), we get: \[ \sqrt{1 + \frac{1}{t}} \approx 1 + \frac{1}{2t} \] Thus, the expression becomes: \[ t \times \left( 1 + \frac{1}{2t} \right) = t + \frac{1}{2} \] So, the value of the limit is: \[ \lim_{t \to \infty} \sqrt{t^2 + t - t} = 0.5 \] Thus, the correct answer is \( \boxed{0.5} \).