","comment":{"@type":"Comment","text":" First find the value for which the function is approaching from the left and right hand side of the number line, that number is the limit value."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 12","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" 7.5","position":2},{"@type":"Answer","encodingFormat":"text/html","text":" 13.01","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 13","position":1,"answerExplanation":{"@type":"Comment","text":" From the left, as the value of the variable $$y$$ approaches $$1$$, $$f(y)$$ approaches 13. From the right, as the value of the variable $$y$$ approaches $$1$$, $$f(y)$$ approaches 13. The function approaches the same value from both directions, so the approx. value of $$\\underset{y\\to 1}{\\mathop{\\lim }}\\,f(y)=13$$.","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Estimating limit values from tables Quiz 1","text":" Find the approx. value of $$\\underset{y\\to \\dfrac{3}{5}}{\\mathop{\\lim }}\\,f(y)$$ such that $$ \\\\ $$","comment":{"@type":"Comment","text":" First find the value for which the function is approaching from the left and right hand side of the number line."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 2.5","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" 0.6","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" 4.5","position":2}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" The limit does not exist","position":3,"answerExplanation":{"@type":"Comment","text":" From the left, as the value of the variable $$y$$ approaches $$0.6$$, $$f(y)$$ approaches $$2.5$$. From the right, as the value of the variable $$y$$ approaches $$0.6$$, $$f(y)$$ approaches $$4.5$$. The function approaches a different value from both directions, so the limit does not exist.","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Estimating limit values from tables Quiz 1","text":" Find the combined limit value of $$\\underset{y\\to 4}{\\mathop{\\lim }}\\,f(y)$$ and $$\\underset{x\\to 5}{\\mathop{\\lim }}\\,f(x)$$, such that $$ \\\\ $$
and
","comment":{"@type":"Comment","text":" First find the value for which the function is approaching from the left and right hand side of the number line, that number is the limit value."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 12","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" 7","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" 11","position":2}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 10","position":3,"answerExplanation":{"@type":"Comment","text":"For $$\\underset{y\\to 4}{\\mathop{\\lim }}\\,f(y)$$: From the left, as the value of the variable $$y$$ approaches $$4$$, $$f(y)$$ approaches $$3$$. From the right, as the value of the variable $$y$$ approaches $$4$$, $$f(y)$$ approaches $$3$$. The function approaches the same value from both directions, so the approx. value of $$\\underset{y\\to 4}{\\mathop{\\lim }}\\,f(y)=3$$. Similarly, $$\\underset{x\\to 5}{\\mathop{\\lim }}\\,f(x)=7$$ Combined value = $$3+7=10$$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Estimating limit values from tables Quiz 1","text":" Find the average of the approx. values of $$\\underset{y\\to 13}{\\mathop{\\lim }}\\,f(y)$$ and $$\\underset{x\\to -12}{\\mathop{\\lim }}\\,f(x)$$, such that $$ \\\\ $$
and
","comment":{"@type":"Comment","text":" First find the value for which the function is approaching from the left and right hand side of the number line, that number is the limit value."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$5$$","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" $$-6.5$$","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" $$2.5$$","position":3}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" $$-2.5$$","position":2,"answerExplanation":{"@type":"Comment","text":" For $$\\underset{y\\to 13}{\\mathop{\\lim }}\\,f(y)$$: From the left, as the value of the variable $$y$$ approaches $$13$$, $$f(y)$$ approaches $$-9$$. From the right, as the value of the variable $$y$$ approaches $$13$$, $$f(y)$$ approaches $$-9$$. The function approaches the same value from both directions, so the approx. value of $$\\underset{y\\to 13}{\\mathop{\\lim }}\\,f(y)=-9$$. Similarly, $$\\underset{x\\to -12}{\\mathop{\\lim }}\\,f(x)=4.$$ Average value = $$\\dfrac{4+(-9)}{2}=\\dfrac{-5}{2}=-2.5$$","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]},{"@type":"Question","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","educationalLevel":"beginner","name":"Estimating limit values from tables Quiz 1","text":" If you are given the following information organized in two tables, then find the approx. value of $${{a}^{2}}+3bd+7c+1$$, where $$\\underset{x\\to a}{\\mathop{\\lim }}\\,f(x)=b$$ and $$\\underset{y\\to c}{\\mathop{\\lim }}\\,f(y)=d$$? $$ \\\\ $$
and $$ \\\\ $$
","comment":{"@type":"Comment","text":" First find the approx. number for which the other numbers are approaching, then put those values in the given expression."},"encodingFormat":"text/markdown","suggestedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 12","position":0},{"@type":"Answer","encodingFormat":"text/html","text":" 54","position":1},{"@type":"Answer","encodingFormat":"text/html","text":" -24","position":2}],"acceptedAnswer":[{"@type":"Answer","encodingFormat":"text/html","text":" 62","position":3,"answerExplanation":{"@type":"Comment","text":" From the first table, logically by seeing the symmetry it can be concluded that the top row approaches column 6, i.e., $$a\\approx 6$$ and the bottom row approaches 11, i.e., $$b\\approx 11$$. From the second table, similarly, it can be concluded that $$c\\approx 13$$ and $$d=-2$$. Therefore, $${{a}^{2}}+3bd+7c+1={{6}^{2}}+3\\times 11\\times (-2)+7\\times 13+1=36-66+91+1=62$$.","encodingFormat":"text/html"},"comment":{"@type":"Comment"}}]}]}