Limits at Infinity Worksheet
This worksheet focuses on evaluating limits of functions as the variable approaches infinity, covering various techniques and function types relevant to Grade 12 Calculus.
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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.
1. Evaluate the following limit:
$\lim_{x \to \infty} \frac{3x^2 - 5x + 2}{2x^2 + 7x - 1}$
2. Evaluate the following limit:
$\lim_{x \to -\infty} \frac{4x^3 + 6x - 1}{x^2 - 3x + 5}$
3. What is the horizontal asymptote of the function $f(x) = \frac{2x - 1}{x + 3}$?
y = 0
y = 1
y = 2
No horizontal asymptote
4. For the function $g(x) = \frac{x^2 + 5}{x^3 - 2x + 1}$, what is $\lim_{x \to \infty} g(x)$?
0
1
$\infty$
Does not exist
5. If the degree of the numerator is less than the degree of the denominator, the limit as $x \to \infty$ is .
6. If the degree of the numerator is greater than the degree of the denominator, the limit as $x \to \infty$ is .
7. For $f(x) = e^{-x}$, $\lim_{x \to \infty} f(x) = 0$.
True
False
8. The function $h(x) = \sin(x)$ has a limit as $x \to \infty$.
True
False
9. Determine the limit: $\lim_{x \to \infty} (\sqrt{x^2 + 4x} - x)$