Limits of functions of two variables practice problems
Limits of functions of two variables: by substitution, by cancelling, or showing two paths disagree. 60 problems with worked solutions; in 60 of them every
equation is proved by a computer algebra system.
19 worked problems
Try each one before opening the solution. Every line is labeled with what checked it. How we verify
Find \( \displaystyle \lim_{(x,y)\to(0,0)} \frac{x y}{x^{2} + y^{2}} \), or show it does not exist.
Approach (0, 0) along the curves y = m*x for different m.
Reviewed
\[ \lim_{x \to 0} 0 = 0 \]
Along y = 0.✓ Proved
\[ \lim_{x \to 0} \frac{1}{2} = \frac{1}{2} \]
Along y = x: a different value, so the limit does not exist.✓ Proved
Answer \( \text{does not exist} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(-3,1)} \frac{x^{2} + 4 x y + 3 y^{2}}{x + 3 y} \), or show it does not exist.
Substitution gives 0/0. Factor the numerator and cancel the common factor (it is nonzero near, but not at, the point).
Reviewed
\[ \left(x + y\right) \left(x + 3 y\right) = x^{2} + 4 x y + 3 y^{2} \]
The numerator factored.✓ Proved
\[ \left. x + y \right|_{\substack{ x=-3\\ y=1 }} = -2 \]
Then substitute.✓ Proved
Answer \( -2 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(2,1)} \frac{x^{2} - x y - 2 y^{2}}{x - 2 y} \), or show it does not exist.
Substitution gives 0/0. Factor the numerator and cancel the common factor (it is nonzero near, but not at, the point).
Reviewed
\[ \left(x - 2 y\right) \left(x + y\right) = x^{2} - x y - 2 y^{2} \]
The numerator factored.✓ Proved
\[ \left. x + y \right|_{\substack{ x=2\\ y=1 }} = 3 \]
Then substitute.✓ Proved
Answer \( 3 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(0,0)} \frac{x^{2} y}{x^{4} + y^{2}} \), or show it does not exist.
Approach (0, 0) along the curves y = m*x**2 for different m.
Reviewed
\[ \lim_{x \to 0} 0 = 0 \]
Along y = 0.✓ Proved
\[ \lim_{x \to 0} \frac{1}{2} = \frac{1}{2} \]
Along y = x**2: a different value, so the limit does not exist.✓ Proved
Answer \( \text{does not exist} \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
\[ \left. x + y \right|_{\substack{ x=1\\ y=1 }} = 2 \]
Then substitute.✓ Proved
Answer \( 2 \)
✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(-2,3)} - y + \cos{\left(x y \right)} \), or show it does not exist.
The function is continuous at the point (no division by zero there), so substitute.
Reviewed
\[ \left. - y + \cos{\left(x y \right)} \right|_{\substack{ x=-2\\ y=3 }} = -3 + \cos{\left(6 \right)} \]
Direct substitution.✓ Proved
Answer \( -3 + \cos{\left(6 \right)} \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(0,3)} - y + \cos{\left(x y \right)} \), or show it does not exist.
The function is continuous at the point (no division by zero there), so substitute.
Reviewed
\[ \left. - y + \cos{\left(x y \right)} \right|_{\substack{ x=0\\ y=3 }} = -2 \]
Direct substitution.✓ Proved
Answer \( -2 \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(2,-2)} - y + \cos{\left(x y \right)} \), or show it does not exist.
The function is continuous at the point (no division by zero there), so substitute.
Reviewed
\[ \left. - y + \cos{\left(x y \right)} \right|_{\substack{ x=2\\ y=-2 }} = \cos{\left(4 \right)} + 2 \]
Direct substitution.✓ Proved
Answer \( \cos{\left(4 \right)} + 2 \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
Find \( \displaystyle \lim_{(x,y)\to(-1,-2)} - y + \cos{\left(x y \right)} \), or show it does not exist.
The function is continuous at the point (no division by zero there), so substitute.
Reviewed
\[ \left. - y + \cos{\left(x y \right)} \right|_{\substack{ x=-1\\ y=-2 }} = \cos{\left(2 \right)} + 2 \]
Direct substitution.✓ Proved
Answer \( \cos{\left(2 \right)} + 2 \)
✓ Nihil obstat Lines: 1 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt ·
How we verify
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