Recurrent Neural Networks are designed to handle sequences by maintaining a memory of past information in their hidden state. This works well for short-term dependencies, like predicting the next word in "the cat sat on the..." where "mat" is a likely guess based on the immediate context.

But what if the context needed is much further back? Consider trying to complete this sentence: "I grew up in France, where the culture is rich and the food is exquisite. I still remember the beautiful summers... That's why I speak fluent..." The key piece of information, "France," appeared long before the blank. A standard RNN struggles to connect these distant pieces of information. This difficulty in handling long-range dependencies is its primary weakness.

To understand why RNNs struggle, we need to look at how they learn. During training, the network calculates an error and then sends a signal, called a gradient, backward through the network to adjust its weights. This process is called backpropagation. In an RNN, this signal must travel back through every time step in the sequence.

Imagine this gradient signal as a message being whispered from one person to the next in a long line. If each person whispers a little quieter than they heard, the message will eventually fade into nothing. This is the vanishing gradient problem. Mathematically, it happens when the gradients are repeatedly multiplied by numbers smaller than 1. After many multiplications, the signal becomes so small that it's effectively zero.

When the gradient vanishes, the network can no longer learn from distant past events. The weights for those early time steps don't get updated, and the network essentially forgets the initial context.

The opposite problem can also occur. What if each person in the line shouts the message louder than they heard it? The message quickly becomes a deafening roar, and the original words are lost. This is the exploding gradient problem.

It happens when gradients are repeatedly multiplied by numbers larger than 1. The signal grows exponentially until it becomes an enormous value, represented as NaN (Not a Number) in computing. This leads to massive, unstable updates to the network's weights, causing the model to fail to learn anything useful. The training process effectively breaks.

Both vanishing and exploding gradients make it incredibly difficult for a simple RNN to reliably learn connections between events that are far apart in a sequence. This limitation was a major barrier until more sophisticated architectures were developed to specifically control the flow of information over time.

These challenges paved the way for new types of recurrent networks, like LSTMs, which were designed to remember information for long periods.