Continuity mistake: In the extended version, where Martha is preparing breakfast, she places a box of Cheerios on the table. Then she looks out the window at Clark. The shot changes to an exterior shot of her looking out the window, but there is a pitcher of milk next to the box of Cheerios that was not there before.
Continuity mistake: While Lex is telling Superman his plans with the missiles at his home, he is touching his cane with both hands then only then both hands back again.
Continuity mistake: While Lois and Superman are being mugged in the alley, Lois kicks the thief and protects herself leaning with her face against the wall and her arms widely spread. A frame later, when Clark grabs the bullet, her arms are close to her head.
Continuity mistake: When Superman lifts the plane during the storm, after the pilot asks what's going on, the copilot glances at the window grabbing its side. From the opposite angle he is away, repeating all movements.
Continuity mistake: In the extended edition, before Superman enters Lex's lair his hair is short, then when he's being shot at, his hair is much longer and bulkier. When he finally enters the place, it's back to short.
Continuity mistake: The two cops are following Otis at Grand Central Station and two girls, one in yellow, pass by their left side. When the angle changes to a closer one, the girls are in the back and pass by again.
Continuity mistake: When Clark shows Perry a white paper with directions of where to send the payroll, the way he holds the paper differs between shots.
Continuity mistake: Lois exits Perry's office and Clark enters to talk about salary. His hat swaps from facing up to down between shots.
Continuity mistake: When Clark is about to jump in front of the train, there's a crossing sign shaped as an X. Depending on the shot the front upper left part is either bent and about to break, straight, or missing.
Continuity mistake: When the Kent's stop the truck and see the cradle, the sky swaps from sunny and clear to heavily cloudy in a second.
Continuity mistake: When Kent's father pulls over his car, he holds the steering wheel with his right hand on the top. From the side, barely a frame before, he has his left hand on the side.
Continuity mistake: The first wide shot of Superman arriving at the broken railway shows no logs hanging. The close shots, and then on, do show logs.
Continuity mistake: The way the bags are positioned in the Kent's pickup differs between the side and back angles.
Continuity mistake: When Lois exits the alley where Clark grabs the bullet, her hair is much shorter than when she entered.
Continuity mistake: When Superman arrives at the ditch by the railway, the gap is 3 times his length. In the next shot the gap has shrunk and now measures exactly Superman's height.
Continuity mistake: In Lex Luthor's lair, the bottles to the left of the TV broadcasting the news disappear in the following shot.
Continuity mistake: When Superman is in Luthor's lair and Lex calls Ms. Teschmaker , the lamp on the floor changes from lit to unlit depending on the shot.
Continuity mistake: When hanging from the bridge, the damage on the school bus' hood is different depending on the shot.
Continuity mistake: When Superman walks to Luthor's lair, the amount of snow on the bridge of the snow-chamber differs between the CCTV shot and the one from the inside.
Continuity mistake: When Ms. Teschmaker is sunbathing, the trolley by her side is close to her thigh. In the wider angle is by her ankle.
Answer: He wanted to see for himself if the stories were true. Some reporters tend to exaggerate the facts and if he had any other weaknesses. He couldn't be sure the kryptonite would work.
While I don't exactly disagree that Lex wanted to see for himself how invincible Superman is, I don't think that's the main reason why he did it. The bullets and fire were simply a charade to make it look like this was his security system. Keep in mind, he wanted Superman to enter his lair because the real trap was the Kryptonite that he had in the lead box.
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