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  • Exercises for Limits and Continuity
    https://math.libretexts.org/Courses/Montana_State_University/M273%3A_Multivariable_Calculus/14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/Limits_and_Continuity/Exercises_for_Limits_and_Continuity
    \(\displaystyle \lim_{(x,y)→(a,b)}\left[\dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)}\right] = \frac{2\left(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\right) - 4 \left(\displaystyle \lim_{(x,y)→(a,b)}g(x,y)\...lim(x,y)→(a,b)[2f(x,y)−4g(x,y)f(x,y)−g(x,y)]=2(lim(x,y)→(a,b)f(x,y))−4(lim(x,y)→(a,b)g(x,y))lim(x,y)→(a,b)f(x,y)−lim(x,y)→(a,b)g(x,y)=2(5)−4(2)5−2=23 4) Show that the limit lim(x,y)→(0,0)5x2yx2+y2 exists and is the same along the paths: y-axis and x-axis, and along y=x.
  • Exercises for Limits and Continuity
    https://math.libretexts.org/Courses/Georgia_State_University_-_Perimeter_College/MATH_2215%3A_Calculus_III/14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/Exercises_for_Limits_and_Continuity
    \(\displaystyle \lim_{(x,y)→(a,b)}\left[\dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)}\right] = \frac{2\left(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\right) - 4 \left(\displaystyle \lim_{(x,y)→(a,b)}g(x,y)\...lim(x,y)→(a,b)[2f(x,y)−4g(x,y)f(x,y)−g(x,y)]=2(lim(x,y)→(a,b)f(x,y))−4(lim(x,y)→(a,b)g(x,y))lim(x,y)→(a,b)f(x,y)−lim(x,y)→(a,b)g(x,y)=2(5)−4(2)5−2=23 4) Show that the limit lim(x,y)→(0,0)5x2yx2+y2 exists and is the same along the paths: y-axis and x-axis, and along y=x.
  • 3.5: Exercises for Limits and Continuity
    https://math.libretexts.org/Courses/Misericordia_University/MTH_226%3A_Calculus_III/Chapter_14%3A_Functions_of_Multiple_Variables_and_Partial_Derivatives/3.05%3A_Exercises_for_Limits_and_Continuity
    \(\displaystyle \lim_{(x,y)→(a,b)}\left[\dfrac{2f(x,y) - 4g(x,y)}{f(x,y) - g(x,y)}\right] = \frac{2\left(\displaystyle \lim_{(x,y)→(a,b)}f(x,y)\right) - 4 \left(\displaystyle \lim_{(x,y)→(a,b)}g(x,y)\...lim(x,y)→(a,b)[2f(x,y)−4g(x,y)f(x,y)−g(x,y)]=2(lim(x,y)→(a,b)f(x,y))−4(lim(x,y)→(a,b)g(x,y))lim(x,y)→(a,b)f(x,y)−lim(x,y)→(a,b)g(x,y)=2(5)−4(2)5−2=23 4) Show that the limit lim(x,y)→(0,0)5x2yx2+y2 exists and is the same along the paths: y-axis and x-axis, and along y=x.

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