FPS Camera Control rotation limits

I am kind of new to Unity and don't have a lot of code experience. I've been working on an FPS game and have written the following code:

 using UnityEngine;
 using System.Collections;
 
 public class fps_control : MonoBehaviour {
     public Rigidbody otherRb;
     public float speed;
     public float turn;
     public float limit_up;
     public float limit_down;
     public GameObject parentObj;
 
     private float rot;
 
     void Awake () {
         Cursor.lockState = CursorLockMode.Locked;
         otherRb = parentObj.GetComponent <Rigidbody> ();
         float xstart = Input.mousePosition.x;
         float ystart = Input.mousePosition.y;
     }
     void FixedUpdate () {
         float xturn = Input.GetAxis("Mouse X");
         float yturn = Input.GetAxis ("Mouse Y");
         parentObj.transform.Rotate (0f, xturn * turn, 0f);
         if (transform.rotation.x <= limit_up) {
             if (transform.rotation.x >=limit_down) {
                 transform.Rotate (-yturn * turn, 0f, 0f);
             }
             else {
                 rot = limit_down;
                 transform.rotation = Quaternion.Euler (rot,0f,0f);
             }
         }
         else {
             rot = limit_up;
             transform.rotation = Quaternion.Euler (rot,0f,0f);
         }
     }
 }

The problem is that the rotation limits I tried to set for the camera aren't working (resulting in the character doing some crazy somersaults). What am I doing wrong? Please help, and thank you to anyone who answers (even if your answer is crap... I still appreciate your effort).