When the learning rate $\eta$ is chosen to be too small, each gradient descent update moves the parameter $x$ only a tiny distance toward the optimum. This results in extremely slow progress, with the algorithm requiring a large number of iterations to reach a satisfactory solution. For instance, applying gradient descent to the quadratic $f(x) = x^2$ with $\eta = 0.05$ and starting from $x = 10$, the parameter value is still approximately $3.49$ after $10$ iterations—far from the optimal solution at $x = 0$. While a small learning rate ensures that the first-order Taylor approximation remains valid and the function value decreases at every step, the practical cost is an unacceptably slow convergence rate.