Understanding Limits and Asymptotes

The video tutorial explores the function f(x) = x / sqrt(x^2 + 1) and its limits as x approaches positive and negative infinity. The instructor simplifies the function by focusing on dominant terms, leading to the conclusion that the limits are 1 and -1, respectively. The tutorial also covers graphing the function, identifying horizontal asymptotes, and plotting key points like f(0). The approach is intuitive, emphasizing understanding over rigorous mathematical proofs.

What is the function f(x) defined as in the video?

• x over x squared

• x plus the square root of x^2+1

• x squared plus 1

• x over the square root of x^2+1

As x becomes very large, which term dominates in the denominator of f(x)?

• x cubed

• x squared

• 1

• x

What does the function f(x) approximate to as x approaches infinity?

• x over x

• x over the absolute value of x

• x over x squared

• x squared over x

What is the value of the limit of f(x) as x approaches positive infinity?

• Infinity

• 0

• 1

• -1

What is the value of the limit of f(x) as x approaches negative infinity?

• -1

• 1

• Infinity

• 0

What is the horizontal asymptote of f(x) as x approaches positive infinity?

• y = -1

• y = 0

• y = x

• y = 1

What is the horizontal asymptote of f(x) as x approaches negative infinity?

• y = x

• y = 0

• y = 1

• y = -1

What is the value of f(0)?

• 0

• 1

• Infinity

• -1

What is the key to determining horizontal asymptotes according to the video?

• Finding the derivative

• Using a graphing calculator

• Identifying dominant terms

• Calculating the integral

What should you do to verify the graph of the function?

• Use a calculator to plot more points

• Guess the shape of the graph

• Ignore the graph

• Use only theoretical calculations