Limits at Infinity Worksheet
Explore limits of functions as x approaches positive or negative infinity with this Grade 10 calculus worksheet.
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Limits at Infinity
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Read each question carefully and determine the limit of the given function as x approaches infinity or negative infinity. Show all your work for full credit.
Evaluate the following limits.
1. Find: \( \lim_{x \to \infty} \frac{3x^2 - 2x + 1}{5x^2 + 4x - 7} \)
2. Find: \( \lim_{x \to -\infty} \frac{x^3 + 2x - 1}{2x^2 + 5x + 3} \)
3. Which of the following functions has a horizontal asymptote at \( y = 0 \) as \( x \to \infty \)?
\( f(x) = \frac{x^2}{x+1} \)
\( f(x) = \frac{2x+1}{x^2-1} \)
\( f(x) = \frac{3x+5}{x-2} \)
\( f(x) = x^3 - 4x \)
4. What is the limit of \( f(x) = e^{-x} \) as \( x \to \infty \)?
\( \infty \)
\( -\infty \)
\( 0 \)
\( 1 \)
5. If the degree of the numerator is less than the degree of the denominator, the limit as \( x \to \pm\infty \) is .
6. If the degree of the numerator is greater than the degree of the denominator, the limit as \( x \to \pm\infty \) is or .
7. Consider the function \( g(x) = \frac{2x^2 + 1}{x^2 - 4} \).
a) Find \( \lim_{x \to \infty} g(x) \) and \( \lim_{x \to -\infty} g(x) \).
b) What does this tell you about the horizontal asymptotes of the function?
c) Sketch the graph of \( g(x) \) on the coordinate plane above, including any horizontal asymptotes.