# Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limit lim_{xrightarrow2}frac{5cdot x^2-20}{x-2}

Question
Limits and continuity
Find the function The following limits represent the slope of a curve y=f(x) at the point {a,f(a)} Determine a possible function f and number a, then calculate the limit
$$\lim_{x\rightarrow2}\frac{5\cdot x^2-20}{x-2}$$

2021-02-27
Consider the given limit $$\lim_{x->2}\frac{5\cdot x^2-20}{x-2}$$
Note that, $$f'(a)=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}$$
That implies, $$f(x)=5x^2$$ and $$a=2$$
Now obtain the limit as follows
$$\lim_{x->2}\frac{5\cdot x^2-20}{x-2}=\lim_{x->2}\frac{5(x^2-4)}{x-2}$$
$$=\lim_{x\rightarrow2}\frac{5(x^2-2^2)}{x-2}$$
$$=\lim_{x\rightarrow2}\frac{5(x-2)(x+2)}{(x-2)}$$
$$=\lim_{x\rightarrow2}5(x+2)$$
$$=5(2+2)$$
$$=5\cdot4$$
$$=20$$
Hence, the value of the given limit is 20.

### Relevant Questions

Consider the curves in the first quadrant that have equationsy=Aexp(7x), where A is a positive constant. Different valuesof A give different curves. The curves form a family,F. Let P=(6,6). Let C be the number of the family Fthat goes through P.
A. Let y=f(x) be the equation of C. Find f(x).
B. Find the slope at P of the tangent to C.
C. A curve D is a perpendicular to C at P. What is the slope of thetangent to D at the point P?
D. Give a formula g(y) for the slope at (x,y) of the member of Fthat goes through (x,y). The formula should not involve A orx.
E. A curve which at each of its points is perpendicular to themember of the family F that goes through that point is called anorthogonal trajectory of F. Each orthogonal trajectory to Fsatisfies the differential equation dy/dx = -1/g(y), where g(y) isthe answer to part D.
Find a function of h(y) such that x=h(y) is the equation of theorthogonal trajectory to F that passes through the point P.
Evaluate the following limits.
$$\lim_{x\rightarrow2}\frac{\sin(x-2)}{x^2-4}$$
Find each of the following limits. If the limit is not finite, indicate or for one- or two-sided limits, as appropriate.
$$\lim_{x\rightarrow\infty}\frac{4x^3-2x-1}{x^2-1}$$
Find the limits. Write $$\infty\ or\ -\infty$$ where appropriate.
$$\lim_{x\rightarrow2^-}\frac{x^2-3x+2}{x^3-2x^2}$$
Find the limits:
$$\lim_{x\rightarrow2}\frac{\sqrt{x^2+12}-4}{x-2}$$
Find the limits:
$$\lim_{x\rightarrow2}\frac{x^2-7x+10}{x-2}$$
$$\lim_{t\rightarrow0}\frac{5t^2}{\cos t-1}$$
Suppose the functions f(x) and g(x) are defined for all x and that $$\lim_{x\rightarrow0}f(x)=\frac{1}{2}$$ and $$\lim_{x\rightarrow0}g(x)=\sqrt2$$. Find the limits as $$x\rightarrow0$$ of the following functions. $$f(x)\frac{\cos x}{x-1}$$
Let $$a_n\rightarrow0$$, and use the Algebraic Limit Theorem to compute each of the following limits (assuming the fractions are always defined):
$$\lim_{n\rightarrow\infty}\frac{1+2a_n}{1+3a_n-4(a_n)^2}$$
$$f_{X}(t)=f_{Y}(t)=\begin{cases}\frac{2}{t^{2}},\ t>2\\0,\ otherwise \end{cases}$$