Estimate the limit of 𝑎ℎ−1 ℎ as ℎ → 0, using technology, for various values of 𝑎 > 0.

A technology tool is used to evaluate \( f(h) = \frac{2.5^h - 1}{h} \) for values of \( h \) close to zero. The results are displayed in the following table. Use the sequence of values to estimate \( \lim_{h \to 0} \frac{2.5^h - 1}{h} \) correct to two decimal places.

Consider the function \(f(h) = \frac{5^h - 1}{h}\). (a) Complete the table below by calculating \(f(h)\) for each given value of \(h\), correct to four decimal places. (2 marks) (b) Use your results from part (a) to estimate \(\displaystyle\lim_{h \to 0} \frac{5^h - 1}{h}\), correct to two decimal places. (1 mark) (c) State the mathematical significance of this limit in relation to the derivative of \(y = 5^x\) at \(x = 0\). (1 mark)

A student uses a calculator to evaluate \(\frac{a^h - 1}{h}\) for \(a = 3\) and different positive values of \(h\) approaching zero. The table shows the results. Which value is the best estimate of \(\lim_{h \to 0} \frac{3^h - 1}{h}\)?