In which sonnet is this poem written? Analyze the poem.

In "Sonnet 43" Elizabeth Barett Browning begins with the rhetorical question "How do I love thee? Let me count the ways."
The sonnet, one the Victorian poet's most famous, seeks to explain to her audience the immense depth and unyielding love she has for him. Instead of just saying that she loves him, she has prepared a list of the different ways that she does. Her love for him is everywhere and never-ending; in fact, her love for him will live on even after she has died.

I love thee to the depth and breadth and heightMy soul can reach, when feeling out of sightFor the ends of Being and ideal Grace.

Here in lines 2–4, Browning uses metaphor to describe the vast expanse of her love. It can go anywhere her soul can go, so her love is never not there, and there are no places her love cannot reach. When he is away from her feeling/ sight her love for him is still there.

I love thee to the level of everyday'sMost quiet need, by sun and candle-light.

Her love for him isn't just a passionate love; it's one that fits into the everyday life. She loves him with a "quiet need" a nice juxtaposition to the passion she's about to describe, and she loves him at all hours of the day "sun and candle-light" is basically her way of telling him she loves him from sunrise and even into the evening after the sun has set.

love thee freely, as men strive for Right;I love thee purely, as they turn from Praise.

She tells him that her love is free and pure. She doesn't list the ways she loves him in order to gain adulation; instead, her love for him is uninhibited and given without expectation. People have a desire to "strive for right" or the morally good. Her love for him is this good.

I love thee with the passion put to useIn my old griefs, and with my childhood's faith.

Here she shows the passion she brings in the way she loves him. She loves him the way a child has faith (again, without limit and accepting) and in the same way, she uses to have in "old griefs" or previous pains. She compares the actions to allow him to see how much she is putting into her love for him.

I love thee with a love I seemed to loseWith my lost saints –
I love thee with the breath,Smiles, tears, of all my life! –

Tese 7th and 8th ways of demonstrating love explain that she loves him with the love she has lost by letting others down. It also reminds him that she loves him with every smile that crosses her faith, but she also loves him in the sad, or teary times.

and, if God choose,I shall but love thee better after death.

And finally, if God chooses to take one of them (because she will not leave him by her volition) she will continue to love him, even more, after death.