Revealing mistake: On the Golden Gate bridge, the beetle car that skidded a split second before appears totally empty when the van crashes against it.
Revealing mistake: Before the Hoover Dam breaks, the places where all the cracks and water will later appear are easily traced on the concrete, covered in plaster and painted. Either that, or the real Hoover Dam has burst many many times and always through the same spots.
Continuity mistake: After Jor-el talks to Von-Dah he places his hands on a man's chest. In the following shot they're on his shoulders.
Deliberate mistake: Before the school bus crashes, a brown car is seen coming. When the angle changes we still see a brown car but it's a totally different model.
Continuity mistake: In the gas station, the "closed" sign falls down but appears up in the next shot (you'll find it in the right side of the screen).
Deliberate mistake: Before Superman traps the black missile, its beam is really big and covers half of the lettering on the side. When Superman is behind, the fire is much smaller and the lettering is visible, allowing us to see Superman.
Continuity mistake: Perry White shows newspapers with day shots of Superman, but he made his appearance at night.
Revealing mistake: When Superman leaves the prison, wires holding him are visible.
Continuity mistake: When the burglar falls from the building, his flashlight is sometimes swinging, but other times it's flat against his body revealing the camera was tilted.
Continuity mistake: When Lois dies, Superman's face is covered in dirt. When he takes off and flies, it's clean.
Continuity mistake: When the gas station falls down, watch the right side: A green car with an open hood suddenly gets replaced by a red car.
Continuity mistake: When the school bus smashes the Golden Gate fence, the aerial shot from the back shows the fence complete, the shot from the front shows it smashed.
Continuity mistake: When Lois dies, the first time Superman sees her the left side of her face has sand and rocks around her cheek. When the angle changes they are at shoulder level.
Continuity mistake: When Superman avoids the train from derailing, the wide shot shows his cape fluttering but the close one not.
Continuity mistake: When Superman moves the school bus away from the void, the wide shot shows a different fence and wreckage, with a long piece of fence missing. The close shot shows a missing space that is slightly bigger than the bus.
Other mistake: On Perry's office, there's a painted sign on the glass that says "Editors Secretary". Being the Daily Planet such an important paper, and with Perry White so picky on grammar mistakes, someone should've noticed that an apostrophe is missing.
Continuity mistake: After Superman breaks Luthor's door, on Lex's table a map is resting on a sculpture on the left. From a back shot, it isn't.
Audio problem: While flying with Superman, Lois falls down and starts to shout. When he grabs her, her mouth closes slightly and her expression changes, but the loud crying continues.
Revealing mistake: When Superman lands the burglar, wires holding him are visible for a brief moment.
Continuity mistake: When Superman saves the wrecked plane, from the outside shots the window has droplets of water, but from the copilot's POV the glass is clean.
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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