Let's (vamos) em vez de Let us (permita-nos) faz mais sentido:
Let's go swimming at my house.
Para fazer uma sugestão informal/um convite, usa-se a contração let's = vamos.
• Let's go to the movies tomorrow. (Vamos ao cinema amanhã.)
• Let's order a pizza tonight. (Vamos pedir um pizza hoje à noite.)
Só para esclarecer que o que perguntei nesta página está respondido por um gramático inglês(W.Stannard Allen) Diz ele que Let's difere de Let us. Let's é usado para fazer um convite, como um imperativo da primeira pessoa do plural: Vamos (nós)!. Let us é uma forma imperativa da segunda pessoa: Deixe-nos (você); Deixa-nos (tu) e não pode ser reduzida para let's). Assim Let us go swim seria Deixe-nos nadar e Let's swim seria Vamos nadar. O problema é que o Duolingo evita as formas reduzidas e Let us talvez seja, para o Dulolingo, Let's. Se o Duolingo pudesse responder...Mas Fercho respondeu abaixo, um ano depois, num comentário bem completo, que inclui let us e let's. Vale a pena ler.
We can use let to mean to rent out a house.
We moved to Australia and let our house in London. A major use of let is for talking about 'allowing' and 'permission'.
My mother let me come. I let her leave early. One pattern is let followed by an OBJECT followed by a BARE INFINITIVE
She let me have the last one. You must let us know when you are coming to New York. I didn't let my friend drive my car. Why did you let Jane do that? Another pattern is let followed by an OBJECT followed by a COMPLEMENT
Let me past. The police aren't letting anybody out of the building. Don't let him get away. Why did you let him in? Notice that we can use reflexive pronouns to talk about what we allow ourselves to do.
I let myself be influenced by her. She let herself out. I didn't let myself be talked into it. Did you let yourself in? Notice that we do not make a contraction out of let us when asking for permission.
Let us help you. Let us pay for this. Let us buy that. Let us take the risk. We use let's when we are making a suggestion.
Let's go to the cinema. Let's stay at home. Let's not argue. (formal) Don't let's argue. (informal) Compare these
Let us watch TV, please. (asking for permission) Let's watch TV. (a suggestion) There are many fixed expressions, idioms and phrasal verbs using let.
Let alone means 'much less'
I've never been to Africa, let alone Ghana. I've never met any actor, let alone Brad Pitt. Let go can mean to dismiss
My company let 20 people go. I was let go three weeks ago. Let your hair down means to lose your inhibitions.
At the office party, everybody let their hair down and we had fun. Don't let it get you down means to stay cheerful although something bad has happened.
Everybody makes mistakes. Don't let it get you down. let you off'/let you off the hook means that you are excused, even though you did something wrong or had something bad to do.
I made a mess but the boss let me off. I was supposed to tidy up after the party but I was let off the hook as Sandra did it. Letting off steam means to get rid of excess energy or frustration.
After work I go to the gym and let off steam on the treadmill. We need to go out after the exams and let off steam. Let me see and let me think are expressions used to give you time to think.
You want a raise? Let me think. Can we discuss this later? A good place for lunch? Let me see. How about the Italian restaurant? Let's say and let's suppose are used to talk about hypothetical situations.
Let's say that you were boss. What would you do? Let's suppose that we don't get the contract. What do we do? Let's hope is used to express a hope.
Let's hope he gets the job. Yes, let's hope so.
let e make não usam o to depois: Let me read the letter = deixe-me ler a carta!! -I want to make you feel as if you want to let me make you tonight" - http://www.vagalume.com.br/ian-gillan-band/you-make-me-feel-so-good.html
Let's = vamos (imperativo da primeira pessoa usado para fazer uma suggestão informal - tornou-se uma expressão fixa.)
• Vamos caminhar após o jantar.
• Let's take a walk after dinner.
Let us = permitir-nos
• Professor, permita-nos sair cedo da aula, por favor.
• Teacher, please let us leave class early.
Creio que se fosse um pedido como todos estão dizendo, deveria haver um ponto de interrogação no final da frase. E eu não me incluo quando quero pedir pra mãe de alguém deixar seu filho ou filha fazer algo. O pedido mais natural seria: Do you let her swim at my house? Já esta frase do Duolingo me parece um convite: Vamos nadar na minha casa.
Escrito assim, também não faz sentido em inglês. "Let's" significa "vamos" e representa a primeira pessoa do imperativo usado para fazer sugestões ou convites informais. "Let + pronome" é usado para dar ou pedir permissão.
• Let's go swimming at my house.
• Vamos nadar na minha casa.
• Please, mom, let us go out with our friends tonight!
• Por favor, mãe, deixe-nos sair com nossos amigos hoje à noite!
• Let me know when you arrive home.
• Avise-me quando você chegar em casa.